A faro shuffle of a deck of playing cards is a shuffle in which the deck is split exactly in half and then the cards in the two halves are perfectly interwoven, such that the original bottom card is still on the bottom and the original top card is still on top.

For example, faro shuffling the list
```python
['ace', 'two', 'three', 'four', 'five', 'six']
```
gives
```python
['ace', 'four', 'two', 'five', 'three', 'six' ]
```

If 8 perfect faro shuffles are performed on a deck of 52 playing cards, the deck is restored to its original order.

Write a function that inputs an integer n and returns an integer representing the number of faro shuffles it takes to restore a deck of n cards to its original order.

Assume n is an even number between 2 and 2000.

```python

USER_FILENAME = "<user_code>"
import sys, json, builtins
import math, heapq, bisect, itertools, functools, collections, statistics, fractions, decimal, operator, array, random, re, string, os, pathlib
try:
    import resource as _res
except Exception:
    _res = None
from collections import Counter, defaultdict, deque, OrderedDict
from functools import lru_cache, reduce
from itertools import accumulate, permutations, combinations, product
from math import gcd, ceil, floor, sqrt
from decimal import Decimal
try:
    from typing import *
except Exception:
    pass
try:
    sys.setrecursionlimit(1_000_000)
except Exception:
    pass
try:
    import site as _site
    try:
        _sys_sites = _site.getsitepackages() if hasattr(_site, "getsitepackages") else []
        for _p in _sys_sites:
            if _p and _p not in sys.path:
                sys.path.append(_p)
    except Exception:
        pass
    try:
        _usp = _site.getusersitepackages()
        if isinstance(_usp, str):
            _usp = [_usp]
        for _p in _usp:
            if _p and _p not in sys.path:
                sys.path.append(_p)
    except Exception:
        pass
except Exception:
    pass

def _disabled_input(*args, **kwargs):
    raise RuntimeError("input() is disabled in callable mode")
builtins.input = _disabled_input

ns = {"__name__": "__judge__"}  # prevent if __name__ == "__main__" blocks
for _k, _v in list(globals().items()):
    if not _k.startswith("_"):
        ns[_k] = _v
_CODE = 'def faro_cycles(deck_size):\n    return [None, 1, 2, 4, 3, 6, 10, 12, 4, 8, 18, 6, 11, 20, 18, 28, 5, 10, 12, 36, 12, 20, 14, 12, 23, 21, 8, 52, 20, 18, 58, 60, 6, 12, 66, 22, 35, 9, 20, 30, 39, 54, 82, 8, 28, 11, 12, 10, 36, 48, 30, 100, 51, 12, 106, 36, 36, 28, 44, 12, 24, 110, 20, 100, 7, 14, 130, 18, 36, 68, 138, 46, 60, 28, 42, 148, 15, 24, 20, 52, 52, 33, 162, 20, 83, 156, 18, 172, 60, 58, 178, 180, 60, 36, 40, 18, 95, 96, 12, 196, 99, 66, 84, 20, 66, 90, 210, 70, 28, 15, 18, 24, 37, 60, 226, 76, 30, 29, 92, 78, 119, 24, 162, 84, 36, 82, 50, 110, 8, 16, 36, 84, 131, 52, 22, 268, 135, 12, 20, 92, 30, 70, 94, 36, 60, 136, 48, 292, 116, 90, 132, 42, 100, 60, 102, 102, 155, 156, 12, 316, 140, 106, 72, 60, 36, 69, 30, 36, 132, 21, 28, 10, 147, 44, 346, 348, 36, 88, 140, 24, 179, 342, 110, 36, 183, 60, 156, 372, 100, 84, 378, 14, 191, 60, 42, 388, 88, 130, 156, 44, 18, 200, 60, 108, 180, 204, 68, 174, 164, 138, 418, 420, 138, 40, 60, 60, 43, 72, 28, 198, 73, 42, 442, 44, 148, 224, 20, 30, 12, 76, 72, 460, 231, 20, 466, 66, 52, 70, 180, 156, 239, 36, 66, 48, 243, 162, 490, 56, 60, 105, 166, 166, 251, 100, 156, 508, 9, 18, 204, 230, 172, 260, 522, 60, 40, 253, 174, 60, 212, 178, 210, 540, 180, 36, 546, 60, 252, 39, 36, 556, 84, 40, 562, 28, 54, 284, 114, 190, 220, 144, 96, 246, 260, 12, 586, 90, 196, 148, 24, 198, 299, 25, 66, 220, 303, 84, 276, 612, 20, 154, 618, 198, 33, 500, 90, 72, 45, 210, 28, 84, 210, 64, 214, 28, 323, 290, 30, 652, 260, 18, 658, 660, 24, 36, 308, 74, 60, 48, 180, 676, 48, 226, 22, 68, 76, 156, 230, 30, 276, 40, 58, 700, 36, 92, 300, 708, 78, 55, 60, 238, 359, 51, 24, 140, 121, 486, 56, 244, 84, 330, 246, 36, 371, 148, 246, 318, 375, 50, 60, 756, 110, 380, 36, 24, 348, 384, 16, 772, 20, 36, 180, 70, 252, 52, 786, 262, 84, 60, 52, 796, 184, 66, 90, 132, 268, 404, 270, 270, 324, 126, 12, 820, 411, 20, 826, 828, 92, 168, 332, 90, 419, 812, 70, 156, 330, 94, 396, 852, 36, 428, 858, 60, 431, 172, 136, 390, 132, 48, 300, 876, 292, 55, 882, 116, 443, 21, 270, 414, 356, 132, 140, 104, 42, 180, 906, 300, 91, 410, 60, 390, 153, 102, 420, 180, 102, 464, 126, 310, 40, 117, 156, 940, 220, 36, 946, 36, 316, 68, 380, 140, 204, 155, 318, 96, 483, 72, 194, 138, 60, 488, 110, 36, 491, 196, 138, 154, 495, 30, 396, 332, 36, 60, 232, 132, 468, 504, 42, 92, 84, 84, 1018, 340, 10, 20, 156, 294, 515, 258, 132, 120, 519, 346, 444, 180, 348, 262, 350, 108, 420, 15, 88, 1060, 531, 140, 240, 356, 24, 252, 140, 358, 492, 253, 342, 60, 543, 330, 1090, 364, 36, 274, 156, 366, 29, 24, 180, 1108, 100, 156, 148, 1116, 372, 522, 1122, 300, 231, 564, 84, 510, 452, 378, 264, 162, 42, 76, 180, 382, 575, 288, 60, 132, 180, 126, 166, 116, 388, 249, 1170, 88, 460, 530, 390, 236, 156, 156, 1186, 140, 44, 298, 476, 18, 180, 300, 200, 24, 280, 60, 516, 1212, 324, 152, 572, 180, 611, 420, 204, 1228, 615, 204, 36, 1236, 174, 72, 140, 164, 28, 156, 138, 534, 100, 418, 1258, 48, 420, 220, 180, 414, 20, 198, 40, 1276, 639, 60, 1282, 16, 60, 161, 1290, 86, 36, 648, 72, 1300, 651, 84, 1306, 120, 198, 300, 524, 146, 659, 60, 126, 260, 221, 442, 1210, 70, 44, 285, 204, 444, 312, 268, 224, 630, 96, 20, 540, 638, 30, 680, 644, 12, 683, 1332, 76, 1372, 100, 216, 588, 1380, 460, 92, 18, 462, 636, 99, 60, 70, 233, 466, 660, 140, 66, 704, 328, 156, 188, 36, 70, 84, 237, 180, 1426, 84, 468, 179, 60, 478, 719, 130, 36, 136, 723, 66, 1450, 1452, 48, 115, 486, 486, 90, 292, 162, 84, 245, 490, 580, 210, 56, 370, 1482, 180, 743, 744, 210, 1492, 132, 166, 1498, 234, 498, 84, 340, 502, 755, 88, 100, 180, 105, 156, 1522, 60, 508, 690, 1530, 18, 204, 364, 54, 66, 771, 204, 24, 1548, 230, 194, 620, 516, 779, 111, 260, 156, 783, 522, 1570, 660, 60, 738, 526, 40, 791, 316, 506, 678, 252, 522, 140, 532, 60, 400, 228, 212, 803, 201, 534, 52, 72, 210, 1618, 1620, 540, 300, 542, 180, 87, 385, 36, 1636, 740, 546, 260, 276, 180, 48, 84, 252, 60, 92, 78, 30, 831, 36, 1666, 1668, 556, 357, 660, 84, 99, 820, 120, 84, 24, 562, 198, 1692, 28, 848, 566, 162, 780, 20, 284, 244, 812, 114, 588, 200, 570, 215, 574, 220, 260, 36, 144, 1732, 692, 96, 828, 1740, 246, 348, 1746, 260, 408, 146, 36, 150, 879, 586, 140, 88, 90, 420, 330, 588, 140, 74, 148, 204, 891, 24, 1786, 596, 198, 810, 716, 598, 48, 25, 50, 684, 276, 198, 362, 252, 220, 429, 424, 606, 911, 180, 84, 290, 305, 276, 732, 830, 612, 393, 144, 60, 923, 602, 154, 72, 156, 618, 780, 1860, 594, 372, 1866, 66, 935, 936, 500, 1876, 939, 90, 804, 84, 72, 472, 60, 90, 756, 135, 210, 1900, 860, 28, 1906, 902, 84, 239, 764, 630, 900, 56, 64, 60, 460, 214, 1930, 644, 84, 444, 276, 646, 924, 388, 290, 1948, 975, 30, 88, 306, 652, 468, 60, 260, 210, 890, 18, 1972, 780, 658, 1978, 282, 660, 44, 1986, 24, 180, 996, 36, 1996, 333][deck_size >> 1]'
_co = compile(_CODE, USER_FILENAME, "exec")
exec(_co, ns, ns)

_target = None
_fn_name = 'faro_cycles'
if _fn_name:
    if "Solution" in ns:
        try:
            _obj = ns["Solution"]()
            if hasattr(_obj, _fn_name):
                _target = getattr(_obj, _fn_name)
        except Exception:
            _target = None
    if _target is None and _fn_name in ns and callable(ns[_fn_name]):
        _target = ns[_fn_name]
# Fallback: if there's exactly one user-defined function, use it
if _target is None:
    _cands = [v for k, v in ns.items() if callable(v) and getattr(v, "__module__", "") in ("__judge__", "__main__")]
    _user_funcs = [c for c in _cands if getattr(c, "__name__", "").startswith("_") is False]
    if len(_user_funcs) == 1:
        _target = _user_funcs[0]
if _target is None:
    raise RuntimeError(f"Could not resolve callable '{_fn_name}'")

_raw = sys.stdin.read().strip()
if _raw:
    _parsed = json.loads(_raw)
    if isinstance(_parsed, dict):
        _kwargs = _parsed.get("kwargs", {})
        _args = _parsed.get("args", [])
    else:
        _args, _kwargs = _parsed, {}
else:
    _args, _kwargs = [], {}
if not isinstance(_args, (list, tuple)):
    _args = [_args]
_res = _target(*_args, **_kwargs)
try:
    sys.stdout.write(repr(_res) + "\n")
except Exception:
    pass

```

Measured memory_bytes: 12696