Sequence the Game

Let $A$ be the following list:

$$ [975, 844, 53, 296, 283, 429, 492, 264, 563, 968, 199, 172, 304, 632, 312, 957, 822, 482, 15, 708, 988, 795, 741, 606, 182, 160, 568, 661, 40, 679, 406, 271, 643, 956, 104, 959, 916, 266, 334, 74, 905, 994, 91, 532, 689, 951, 17, 865, 981, 251, 570, 418, 341, 234, 688, 946, 435, 28, 23, 738, 992, 618, 234, 310, 709, 637, 143, 390, 757, 120, 476, 862, 193, 302, 879, 615, 283, 138, 938, 478, 335, 489, 76, 25, 388, 677, 753, 464, 930, 521, 378, 677, 744, 9, 189, 913, 221, 28, 117, 540, 828, 157, 519, 422, 877, 176, 969, 771, 49, 195, 473, 765, 70, 523, 984, 363, 527, 550, 417, 151, 662, 126, 923, 16, 742, 778, 64, 903, 765, 349, 88, 711, 405, 658, 545, 246, 84, 796, 719, 51, 20, 20, 11, 29, 834, 558, 846, 321, 634, 914, 576, 617, 656, 827, 931, 188, 919, 415, 202, 980, 351, 191, 752, 812, 633, 121, 586, 639, 466, 746, 154, 419, 897, 30, 129, 764, 245, 433, 472, 505, 804, 570, 906, 787, 317, 626, 619, 96, 868, 664, 549, 953, 372, 793, 885, 324, 32, 77, 315, 964, 758, 612, 514, 144, 968, 308, 330, 635, 161, 442, 244, 451, 696, 265, 246, 319, 276, 868, 742, 803, 348, 358, 853, 745, 811, 159, 556, 843, 272, 685, 862, 982, 210, 808, 728, 674, 38, 602, 245, 841, 831, 793, 599, 317, 715, 145, 442, 631, 658, 43, 259, 478, 168, 855, 344, 826, 578, 241, 380, 13, 129, 581, 400, 861, 850, 841, 727, 190, 497, 61, 631, 605, 967, 640, 537, 955, 178, 218, 500, 584, 529, 722, 401, 280, 831, 856, 984, 694, 356, 539, 113, 229, 732, 814, 220, 457, 836, 516, 352, 786, 580, 873, 508, 737, 485, 135, 506, 629, 939, 648, 116, 859, 588, 313, 443, 857, 35, 667, 239, 454, 754, 55, 781, 43, 730, 736, 835, 544, 826, 149, 649, 851, 602, 509, 226, 48, 80, 76, 267, 895, 423, 154, 596, 322, 501, 335, 97, 867, 8, 692, 917, 291, 712, 544, 432, 728, 273, 295, 642, 261, 977, 340, 468, 10, 430, 726, 427, 165, 502, 567, 123, 115, 595, 970, 782, 61, 277, 884, 864, 269, 830, 6, 361, 959, 63, 926, 213, 361, 714, 440, 721, 183, 142, 223, 289, 427, 652, 450, 481, 870, 456, 13, 727, 710, 874, 247, 800, 345, 923, 297, 525, 839, 341, 934, 375, 250, 673, 448, 876, 389, 821, 514, 710, 329, 972, 794, 978, 954, 487, 988, 360, 692, 745, 404, 764, 323, 443, 594, 82, 665, 441, 629, 153, 589, 191, 641, 396, 355, 81, 73, 137, 760, 116, 700, 807, 85, 371, 901, 774, 343, 94, 305, 686, 152, 227, 396, 467, 590, 398, 746, 289, 554, 50, 278, 954, 844, 139, 996, 926, 853, 231, 385, 357, 651, 674, 499, 761, 39, 937, 643, 326, 293, 408, 485, 309, 286, 983, 354, 687, 734, 609, 663, 263, 421, 541, 928, 365, 731, 141, 470, 773, 395, 679, 445, 958, 258, 58, 353, 832, 545, 592, 990, 603, 552, 498, 264, 238, 82, 820, 517, 50, 549, 448, 929, 986, 203, 557, 974, 232, 759, 636, 712, 357, 284, 57, 788, 672, 515, 717, 797, 648, 104, 330, 816, 975, 252, 4, 31, 821, 945, 86, 78, 673, 668, 1, 762, 810, 517, 967, 19, 763, 776, 429, 607, 126, 912, 682, 381, 75, 842, 998, 946, 236, 325, 322, 593, 704, 7, 453, 774, 819, 904, 884, 562, 667, 122, 66, 693, 257, 979, 996, 947, 898, 963, 149, 498, 62, 52, 405, 300, 735, 530, 463, 177, 691, 697, 564, 636, 729, 848, 382, 367, 780, 260, 196, 893, 18, 676, 461, 269, 573, 292, 426, 940, 196, 488, 546, 457, 455, 965, 767, 573, 204, 303, 32, 475, 553, 649, 543, 90, 52, 33, 505, 911, 3, 495, 699, 462, 775, 411, 659, 420, 447, 108, 319, 138, 26, 83, 465, 938, 227, 686, 380, 428, 130, 195, 152, 127, 42, 224, 952, 524, 219, 747, 997, 339, 373, 581, 292, 344, 617, 466, 502, 608, 583, 240, 896, 5, 694, 110, 820, 906, 522, 242, 949, 198, 301, 164, 817, 124, 941, 956, 919, 262, 230, 915, 318, 908, 738, 703, 842, 600, 124, 937, 198, 762, 846, 173, 504, 709, 56, 670, 721, 942, 995, 531, 777, 560, 434, 77, 905, 111, 351, 657, 347, 365, 575, 179, 273, 368, 163, 778, 650, 559, 577, 503, 864, 890, 343, 215, 560, 980, 366, 41, 481, 268, 812, 29, 414, 393, 903, 484, 701, 441, 902, 24, 612, 799, 965, 785, 887, 669, 845, 622, 512, 691, 170, 825, 35, 449, 131, 371, 25, 342, 924, 252, 995, 184, 120, 398, 972, 40, 370, 186, 518, 436, 932, 552, 310, 775, 695, 140, 507, 886, 49, 327, 894, 346, 739, 375, 974, 548, 409, 802, 750, 122, 132, 72, 611, 315, 555, 329, 601, 840, 973, 535, 918, 379, 576, 592, 163, 167, 610, 813, 669, 299, 306, 384, 280, 832, 401, 274, 854, 178, 936, 9, 867, 909, 815, 885, 510, 939, 166, 766, 829, 235, 513, 202, 724, 743, 947, 644, 571, 953, 169, 632, 103, 494, 109, 1, 381, 539, 675, 171, 543, 463, 586, 385, 167, 51, 108, 413, 490, 645, 538, 990, 683, 494, 256, 323, 324, 433, 460, 237, 175, 0, 488, 578, 784, 706, 125, 3, 59, 101, 913, 37, 678, 345, 382, 456, 342, 755, 408, 224, 665, 233, 339, 132, 892, 284, 318, 567, 36, 207, 256, 96, 162, 258, 270, 326, 188, 960, 92, 807, 788, 133, 66, 619, 951, 374, 855, 255, 744, 491, 328, 881, 579, 866, 168, 490, 540, 670, 181, 660, 902, 836, 390, 348, 363, 623, 823, 925, 849, 53, 681, 706, 67, 547, 763, 565, 805, 211, 522, 871, 119, 209, 779, 776, 230, 751, 376, 287, 684, 961, 253, 866, 683, 920, 300, 391, 625, 921, 499, 598, 460, 474, 387, 883, 985, 127, 221, 535, 915, 453, 431, 229, 979, 641, 63, 910, 723, 446, 515, 568, 314, 172, 663, 839, 526, 981, 60, 789, 783, 479, 680, 54, 829, 696, 837, 933, 364, 882, 379, 971, 449, 24, 207, 730, 155, 948, 551, 72, 70, 622, 170, 376, 858, 787, 285, 194, 4, 682, 26, 837, 136, 197, 671, 881, 646, 469, 931, 190, 752, 97, 510, 863, 613, 214, 794, 286, 158, 388, 701, 337, 166, 94, 697, 633, 402, 465, 851, 624, 212, 591, 786, 999, 708, 687, 732, 922, 892, 520, 955, 235, 359, 217, 895, 579, 508, 656, 287, 288, 118, 715, 982, 395, 983, 98, 411, 337, 303, 187, 215, 480, 242, 69, 59, 814, 879, 551, 559, 105, 487, 845, 277, 46, 311, 869, 637, 601, 491, 628, 471, 12, 437, 352, 22, 378, 838, 608, 848, 809, 332, 733, 477, 574, 899, 366, 966, 210, 89, 62, 962, 914, 298, 101, 852, 171, 500, 403, 328, 973, 613, 962, 672, 986, 208, 444, 653, 237, 751, 989, 587, 644, 373, 901, 213, 407, 777, 270, 425, 71, 243, 406, 413, 89, 987, 211, 450, 609, 272, 599, 257, 987, 232, 891, 748, 206, 806, 30, 244, 275, 354, 582, 614, 880, 464, 925, 518, 33, 187, 79, 562, 878, 992, 397, 222, 424, 927, 958, 912, 369, 290, 130, 421, 372, 185, 525, 164, 861, 896, 439, 624, 165, 634, 119, 927, 158, 295, 193, 690, 436, 798, 305, 758, 760, 616, 941, 528, 800, 770, 349, 12, 426, 471, 393, 241, 700, 759, 519, 99, 383, 511, 589, 55, 238, 23, 360, 749, 802, 653, 803, 541, 651, 639, 818, 550, 299, 439, 705, 933, 31, 799, 99, 703, 67, 533, 438, 131, 364, 2, 177, 125, 304, 56, 754, 886, 80, 876, 781, 0, 14, 275, 205, 888, 702, 228, 387, 377, 907, 921, 583, 184, 279, 769, 338, 724, 630, 451, 504, 790, 391, 71, 792, 716, 640, 217, 205, 203, 243, 796, 819, 489, 327, 263, 645, 36, 780, 603, 528, 575, 860, 668, 530, 538, 356, 869, 98, 423, 268, 495, 880, 74, 598, 815, 707, 150, 811, 175, 254, 218, 370, 725, 684, 135, 261, 935, 509, 148, 725, 547, 362, 964, 250, 722, 216, 769, 594, 666, 112, 587, 410, 95, 804, 128, 944, 179, 600, 626, 647, 638, 606, 251, 944, 156, 849, 320, 989, 945, 194, 399, 871, 186, 889, 878, 795, 898, 660, 93, 718, 100, 321, 548, 155, 249, 716, 114, 943, 128, 316, 45, 899, 711, 174, 888, 714, 461, 770, 140, 84, 86, 698, 976, 416, 27, 147, 133, 635, 19, 833, 610, 952, 997, 428, 856, 455, 308, 477, 768, 484, 470, 147, 537, 963, 833, 480, 386, 554, 233, 623, 858, 431, 45, 805, 806, 555, 338, 627, 731, 68, 904, 942, 350, 47, 521, 368, 630, 422, 83, 948, 689, 834, 87, 985, 512, 736, 389, 693, 566, 468, 225, 994, 920, 459, 835, 704, 156, 314, 5, 102, 882, 720, 367, 699, 936, 445, 771, 874, 117, 950, 440, 325, 747, 103, 993, 259, 773, 566, 970, 139, 726, 145, 144, 307, 377, 748, 659, 173, 282, 571, 496, 621, 523, 17, 655, 347, 924, 593, 60, 212, 757, 569, 493, 462, 893, 397, 749, 911, 647, 452, 417, 447, 847, 588, 350, 558, 274, 492, 57, 81, 415, 150, 536, 556, 21, 615, 675, 792, 331, 2, 472, 185, 95, 705, 479, 497, 254, 857, 262, 14, 383, 483, 290, 798, 201, 614, 278, 430, 790, 527, 115, 590, 359, 253, 394, 595, 282, 753, 208, 828, 239, 584, 907, 302, 7, 336, 483, 887, 419, 976, 454, 200, 434, 21, 384, 909, 616, 801, 414, 174, 404, 873, 151, 720, 34, 809, 520, 870, 493, 542, 38, 685, 546, 44, 625, 22, 452, 676, 34, 236, 291, 713, 11, 977, 192, 219, 69, 255, 740, 957, 854, 654, 789, 410, 226, 735, 681, 112, 105, 157, 740, 572, 597, 661, 524, 565, 469, 340, 425, 458, 702, 908, 106, 717, 333, 932, 220, 934, 475, 813, 698, 482, 840, 564, 437, 294, 247, 620, 496, 467, 121, 991, 143, 107, 332, 181, 935, 37, 92, 533, 847, 516, 978, 362, 971, 783, 209, 642, 146, 420, 969, 707, 695, 402, 621, 334, 574, 961, 146, 513, 297, 816, 458, 114, 48, 180, 412, 768, 222, 16, 15, 557, 473, 141, 582, 418, 248, 392, 44, 719, 723, 78, 192, 797, 655, 301, 737, 918, 526, 511, 755, 585, 650, 403, 416, 604, 271, 294, 756, 596, 41, 331, 65, 966, 940, 88, 110, 627, 943, 68, 850, 285, 860, 654, 662, 580, 680, 910, 148, 214, 822, 231, 206, 85, 438, 569, 713, 316, 657, 248, 336, 750, 113, 169, 424, 87, 374, 353, 930, 772, 176, 267, 877, 228, 355, 605, 756, 761, 320, 859, 531, 690, 561, 929, 604, 199, 949, 134, 249, 159, 392, 928, 741, 563, 784, 27, 922, 18, 279, 843, 137, 46, 900, 638, 47, 960, 307, 160, 58, 73, 591, 91, 75, 532, 109, 607, 628, 161, 734, 412, 42, 435, 785, 476, 900, 620, 183, 801, 830, 825, 733, 400, 142, 818, 897, 65, 779, 585, 260, 577, 889, 743, 666, 223, 182, 200, 298, 597, 767, 79, 64, 134, 197, 618, 950, 90, 875, 39, 891, 10, 791, 276, 266, 534, 446, 100, 444, 536, 791, 507, 810, 107, 838, 817, 890, 917, 54, 6, 432, 459, 265, 201, 288, 739, 852, 486, 281, 916, 204, 111, 503, 306, 102, 153, 118, 824, 346, 501, 106, 333, 865, 993, 162, 611, 542, 225, 875, 872, 506, 8, 827, 553, 312, 766, 863, 664, 409, 296, 824, 678, 216, 407, 823, 386, 572, 529, 999, 293, 688, 808, 93, 872, 718, 646, 240, 772, 486, 991, 180, 652, 123, 894, 534, 281, 189, 561, 313, 369, 136, 394, 358, 474, 311, 729, 399, 883, 998, 782, 309, 671, 999000] $$

Let $a_i$ be the $i$th element of $A$, and let $n$ be the length of $A$.

Evaluate $$ \frac{\int_{0}^{2\pi} \left( \prod_{i=1}^{n} \cos(a_i \theta) \right) d\theta}{2\pi} $$

Give the exact answer in LaTeX form.

Submit unofficially

The contest ended at 8:45 p.m. AEST on Tuesday 7th April. All submissions will be unofficial officially.