Your car starts at position 0 and speed +1 on an infinite number line.  (Your car can go into negative positions.)
Your car drives automatically according to a sequence of instructions A (accelerate) and R (reverse).
When you get an instruction "A", your car does the following: position += speed, speed *= 2.
When you get an instruction "R", your car does the following: if your speed is positive then speed = -1 , otherwise speed = 1.  (Your position stays the same.)
For example, after commands "AAR", your car goes to positions 0->1->3->3, and your speed goes to 1->2->4->-1.
Now for some target position, say the length of the shortest sequence of instructions to get there.
Example 1:
Input: 
target = 3
Output: 2
Explanation: 
The shortest instruction sequence is "AA".
Your position goes from 0->1->3.

Example 2:
Input: 
target = 6
Output: 5
Explanation: 
The shortest instruction sequence is "AAARA".
Your position goes from 0->1->3->7->7->6.

 
Note: 

1 <= target <= 10000.

```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 = 'class Solution:\n    def __init__(self):\n        self._dp = {0:0}\n\n    def racecar(self, target):\n        # 车子初始位置0，速度1\n        # 遇到A，position += speed并且speed *= 2\n        # 遇到R，speed = -1如果当前speed是1；speed = 1如果\n        # 给定最终的位置target，问最短的sequence（由A和R组成）能够到达target\n        if target == 0:\n            return 0\n        if target == 1:\n            return 1\n        if target == 2:\n            return 4\n        # 如果一直踩油门的话(连续3个A)\n        # 则会到达0 -> 1 -> 3 -> 7\n        #         k=1  k=2  k=3\n        dp = [2 ** 31 - 1] * (target + 1)\n        dp[0], dp[1], dp[2] = 0, 1, 4\n        for t in range(3, target + 1):\n            k = t.bit_length()\n            if t == 2 ** k - 1:\n                # t = 7, 二进制111, k = 3\n                # 此时表明踩3脚油门就能到了\n                dp[t] = k\n                continue\n            \n            # 情况1：走2 ** (k - 1)的距离\n            # 然后用一个R和m个A去往回走\n            # 这时候已经走了k - 1 + 1 + m + 1步（即A**(k - 1)RA**(m)R步)\n            # 然后去dp里找剩余的步数t - 2 ** (k - 1) + 2 ** m\n            for m in range(k - 1):\n                # 注意：后面的k - 1 + m + 2是当前走了这么多步\n                dp[t] = min(dp[t], dp[t - 2 ** (k - 1) + 2 ** m] + k - 1 + m + 2)\n            \n            # 情况2：直接2 ** k步\n            #（已经走了最大的k + 1步, 就是在最后min函数括号里加的那一部分，加1是指一个R）\n            # 然后再往回走\n            # 走了k + 1步距离就是2 ** k - 1，这个距离已经超越了t\n            # 所以还要继续走2 ** k - 1 - t步才能走到t\n            if 2 ** k - 1 - t > 0:\n                dp[t] = min(dp[t], dp[2 ** k - 1 - t] + k + 1)\n        return dp[target]\n        \n'
_co = compile(_CODE, USER_FILENAME, "exec")
exec(_co, ns, ns)

_target = None
_fn_name = 'racecar'
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: 5830