<p>You are given two <strong>positive</strong> integers <code>n</code> and <code>k</code>.</p>

<p>An integer <code>x</code> is called <strong>k-palindromic</strong> if:</p>

<ul>
	<li><code>x</code> is a <span data-keyword="palindrome-integer">palindrome</span>.</li>
	<li><code>x</code> is divisible by <code>k</code>.</li>
</ul>

<p>Return the<strong> largest</strong> integer having <code>n</code> digits (as a string) that is <strong>k-palindromic</strong>.</p>

<p><strong>Note</strong> that the integer must <strong>not</strong> have leading zeros.</p>

<p>&nbsp;</p>
<p><strong class="example">Example 1:</strong></p>

<div class="example-block">
<p><strong>Input:</strong> <span class="example-io">n = 3, k = 5</span></p>

<p><strong>Output:</strong> <span class="example-io">&quot;595&quot;</span></p>

<p><strong>Explanation:</strong></p>

<p>595 is the largest k-palindromic integer with 3 digits.</p>
</div>

<p><strong class="example">Example 2:</strong></p>

<div class="example-block">
<p><strong>Input:</strong> <span class="example-io">n = 1, k = 4</span></p>

<p><strong>Output:</strong> <span class="example-io">&quot;8&quot;</span></p>

<p><strong>Explanation:</strong></p>

<p>4 and 8 are the only k-palindromic integers with 1 digit.</p>
</div>

<p><strong class="example">Example 3:</strong></p>

<div class="example-block">
<p><strong>Input:</strong> <span class="example-io">n = 5, k = 6</span></p>

<p><strong>Output:</strong> <span class="example-io">&quot;89898&quot;</span></p>
</div>

<p>&nbsp;</p>
<p><strong>Constraints:</strong></p>

<ul>
	<li><code>1 &lt;= n &lt;= 10<sup>5</sup></code></li>
	<li><code>1 &lt;= k &lt;= 9</code></li>
</ul>

class Solution:
    def largestPalindrome(self, n: int, k: int) -> str:
        if k == 1: return '9' * n
        candidate = [0] * n
        power = ([0] * (n-1)) + [1]
        for i in range(n - 2, -1, -1):
            power[i] = (power[i+1] * 10) % k
        @cache
        def all_palindromes(index, div):
            mirror_index = n - 1 - index
            if index > mirror_index:
                return div % k == 0
            for digit in range(9, -1, -1):
                if index == 0 and digit == 0:
                    continue
                candidate[index], candidate[mirror_index] = digit, digit
                next_div = div
                if index == mirror_index:
                    next_div += digit * (power[index]) % k
                else:
                    next_div += digit * (power[index] + power[mirror_index]) % k
                if all_palindromes(index + 1, next_div % k):
                    return True
            return False

        if all_palindromes(0, 0):
            # print('power', power)
            # print('candidate', candidate)
            return "".join(map(str, candidate))
        return ""
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