<p>You are given two numeric strings <code>num1</code> and <code>num2</code> and two integers <code>max_sum</code> and <code>min_sum</code>. We denote an integer <code>x</code> to be <em>good</em> if:</p>

<ul>
	<li><code>num1 &lt;= x &lt;= num2</code></li>
	<li><code>min_sum &lt;= digit_sum(x) &lt;= max_sum</code>.</li>
</ul>

<p>Return <em>the number of good integers</em>. Since the answer may be large, return it modulo <code>10<sup>9</sup> + 7</code>.</p>

<p>Note that <code>digit_sum(x)</code> denotes the sum of the digits of <code>x</code>.</p>

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

<pre>
<strong>Input:</strong> num1 = &quot;1&quot;, num2 = &quot;12&quot;, <code>min_sum</code> = 1, max_sum = 8
<strong>Output:</strong> 11
<strong>Explanation:</strong> There are 11 integers whose sum of digits lies between 1 and 8 are 1,2,3,4,5,6,7,8,10,11, and 12. Thus, we return 11.
</pre>

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

<pre>
<strong>Input:</strong> num1 = &quot;1&quot;, num2 = &quot;5&quot;, <code>min_sum</code> = 1, max_sum = 5
<strong>Output:</strong> 5
<strong>Explanation:</strong> The 5 integers whose sum of digits lies between 1 and 5 are 1,2,3,4, and 5. Thus, we return 5.
</pre>

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

<ul>
	<li><code>1 &lt;= num1 &lt;= num2 &lt;= 10<sup>22</sup></code></li>
	<li><code>1 &lt;= min_sum &lt;= max_sum &lt;= 400</code></li>
</ul>

MOD = int(1e9 + 7)

class Solution:
    @cache
    def countStrings(self, i: int, tight1: bool, tight2: bool, sum_: int) -> int:
        if sum_ < 0: return 0
        if i == len(self.num2): return 1

        lo = int(self.num1[i]) if tight1 else 0
        hi = int(self.num2[i]) if tight2 else 9

        count = 0
        for num in range(lo, hi + 1):
            count = (count % MOD + self.countStrings(
                        i + 1,
                        tight1 and (num == lo),
                        tight2 and (num == hi),
                        sum_ - num) % MOD) % MOD

        return count

    def count(self, num1: str, num2: str, min_sum: int, max_sum: int) -> int:
        self.dp = [[[[ -1 for _ in range(401)] for _ in range(2)] for _ in range(2)] for _ in range(23)]
        leadingZeroes = len(num2) - len(num1)
        num1extended = '0' * leadingZeroes + num1 # If the numbers are 1 and 999, convert 1 into 001
        
        self.num1 = num1extended
        self.num2 = num2
        total = self.countStrings(0, True, True, max_sum)
        unnecessary = self.countStrings(0, True, True, min_sum - 1)
        ans = (total - unnecessary + MOD) % MOD
        return ans

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