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 Solution1:\n    \n    def __init__(self):\n        self.dp = {0: 0}\n    \n    def racecar(self, target: int) -> int:\n        if target in self.dp:\n            return self.dp[target]\n        n = target.bit_length()\n        if 2 ** n - 1 == target:\n            self.dp[target] = n\n        else:\n            self.dp[target] = self.racecar(2 ** n - 1 - target) + n + 1\n            for m in range(n - 1):\n                self.dp[target] = min(self.dp[target], \n                                 self.racecar(target - 2 ** (n - 1) + 2 ** m) + n + m + 1)\n                \n        return self.dp[target]\n            \n        \nclass Solution2:\n    def racecar(self, target: int) -> int:\n    # #analysis + DP\n        memo = [-1] * (target * 2)\n        def DP(s):\n            if memo[s] > 0: return memo[s]\n            duishu = math.log2(s+1)\n            if int(duishu) == duishu: \n                memo[s] = duishu\n                return duishu\n            left = int(duishu)\n            right = int(duishu) + 1\n            memo[s] = right + 1 + DP(2**right - s - 1)\n            for i in range(left):\n                huizou = 2**i - 1\n                memo[s] = min(memo[s], left + i + 2 + DP(s - 2**left + 1 + huizou))\n            return memo[s]\n        return int(DP(target))\n        \n    \n    \n    \n    \nclass Solution:\n    def racecar(self, target: int) -> int:    \n    #classic BFS\n        visited = {(0,1)}\n        queue = collections.deque()\n        queue.append((0,1))\n        steps = 0\n        while queue:\n            size = len(queue)\n            for _ in range(size):\n                s, v = queue.popleft()\n                if s == target: return steps\n                # 'A'\n                cur_s = s + v\n                cur_v = v * 2\n                if (cur_s, cur_v) not in visited and s >= 0:\n                    queue.append((cur_s, cur_v))\n                    visited.add((cur_s, cur_v))\n                #'R'\n                cur_s = s\n                cur_v = -1 if v > 0 else 1\n                if (cur_s, cur_v) not in visited and s >= 0:\n                    queue.append((cur_s, cur_v))\n                    visited.add((cur_s, cur_v))\n            steps += 1"
_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: 7286