# Task

You are given an array `a` of positive integers a. You may choose some integer `X` and update `a` several times, where to update means to perform the following operations:
```
pick a contiguous subarray of length not greater than the given k;
replace all elements in the picked subarray with the chosen X.
```
What is the minimum number of updates required to make all the elements of the array the same?

# Example

For `a = [1, 2, 2, 1, 2, 1, 2, 2, 2, 1, 1, 1] and k = 2`, the output should be `4`.

Here's how a will look like after each update:
```
[1, 2, 2, 1, 2, 1, 2, 2, 2, 1, 1, 1] ->
[1, 1, 1, 1, 2, 1, 2, 2, 2, 1, 1, 1] ->
[1, 1, 1, 1, 1, 1, 2, 2, 2, 1, 1, 1] ->
[1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1] ->
[1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1]
```


# Input/Output


 - `[input]` integer array `a`

  An array of positive integers.

  Constraints:
  
  `10 ≤ a.length ≤ 50,`
  
  `1 ≤ a[i] ≤ 10.`
  

 - `[input]` integer `k`

  A positive integer, the maximum length of a subarray.

  Constraints: `2 ≤ k ≤ 9.`


 - `[output]` an integer

  The minimum number of updates.

```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 array_equalization(a, k):\n    totals, ends = {}, {}\n    for i, n in enumerate(a):\n        if n not in ends: totals[n], ends[n] = 0, -1\n        if i < ends[n]: continue\n        count = (i - ends[n] - 1 + k - 1) // k\n        totals[n] += count\n        ends[n] = max(i, ends[n] + count * k)\n    return min(t + (len(a) - ends[n] - 1 + k - 1) // k\n               for n, t in totals.items() if ends[n] < len(a))'
_co = compile(_CODE, USER_FILENAME, "exec")
exec(_co, ns, ns)

_target = None
_fn_name = 'array_equalization'
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: 6680