<p>You are given three integers <code>start</code>, <code>finish</code>, and <code>limit</code>. You are also given a <strong>0-indexed</strong> string <code>s</code> representing a <strong>positive</strong> integer.</p>

<p>A <strong>positive</strong> integer <code>x</code> is called <strong>powerful</strong> if it ends with <code>s</code> (in other words, <code>s</code> is a <strong>suffix</strong> of <code>x</code>) and each digit in <code>x</code> is at most <code>limit</code>.</p>

<p>Return <em>the <strong>total</strong> number of powerful integers in the range</em> <code>[start..finish]</code>.</p>

<p>A string <code>x</code> is a suffix of a string <code>y</code> if and only if <code>x</code> is a substring of <code>y</code> that starts from some index (<strong>including </strong><code>0</code>) in <code>y</code> and extends to the index <code>y.length - 1</code>. For example, <code>25</code> is a suffix of <code>5125</code> whereas <code>512</code> is not.</p>

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

<pre>
<strong>Input:</strong> start = 1, finish = 6000, limit = 4, s = &quot;124&quot;
<strong>Output:</strong> 5
<strong>Explanation:</strong> The powerful integers in the range [1..6000] are 124, 1124, 2124, 3124, and, 4124. All these integers have each digit &lt;= 4, and &quot;124&quot; as a suffix. Note that 5124 is not a powerful integer because the first digit is 5 which is greater than 4.
It can be shown that there are only 5 powerful integers in this range.
</pre>

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

<pre>
<strong>Input:</strong> start = 15, finish = 215, limit = 6, s = &quot;10&quot;
<strong>Output:</strong> 2
<strong>Explanation:</strong> The powerful integers in the range [15..215] are 110 and 210. All these integers have each digit &lt;= 6, and &quot;10&quot; as a suffix.
It can be shown that there are only 2 powerful integers in this range.
</pre>

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

<pre>
<strong>Input:</strong> start = 1000, finish = 2000, limit = 4, s = &quot;3000&quot;
<strong>Output:</strong> 0
<strong>Explanation:</strong> All integers in the range [1000..2000] are smaller than 3000, hence &quot;3000&quot; cannot be a suffix of any integer in this range.
</pre>

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

<ul>
	<li><code>1 &lt;= start &lt;= finish &lt;= 10<sup>15</sup></code></li>
	<li><code>1 &lt;= limit &lt;= 9</code></li>
	<li><code>1 &lt;= s.length &lt;= floor(log<sub>10</sub>(finish)) + 1</code></li>
	<li><code>s</code> only consists of numeric digits which are at most <code>limit</code>.</li>
	<li><code>s</code> does not have leading zeros.</li>
</ul>

class Solution:
    def numberOfPowerfulInt(self, start: int, finish: int, limit: int, s: str) -> int:
        @cache
        def dp(i,tight,max_num):
            remaining_ln=len(max_num)-i
            if remaining_ln<len(s):return 0
            if remaining_ln==len(s):
                if tight:return int(s)<=int(max_num[i:])
                return 1
            res=0
            if tight and int(max_num[i]) <= limit:
                res+=dp(i+1,True,max_num)
                if int(max_num[i])>0:
                    res+=int(max_num[i])*dp(i+1,False,max_num)
            else:
                res+=(limit+1)*dp(i+1,False,max_num)

            return res                
            
        
        return dp(0,True,str(finish)) - dp(0,True,str(start-1))
59