Find the Original Typed String II
<p>Alice is attempting to type a specific string on her computer. However, she tends to be clumsy and <strong>may</strong> press a key for too long, resulting in a character being typed <strong>multiple</strong> times.</p>

<p>You are given a string <code>word</code>, which represents the <strong>final</strong> output displayed on Alice&#39;s screen. You are also given a <strong>positive</strong> integer <code>k</code>.</p>

<p>Return the total number of <em>possible</em> original strings that Alice <em>might</em> have intended to type, if she was trying to type a string of size <strong>at least</strong> <code>k</code>.</p>

<p>Since the answer may be very large, return it <strong>modulo</strong> <code>10<sup>9</sup> + 7</code>.</p>

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

<div class="example-block">
<p><strong>Input:</strong> <span class="example-io">word = &quot;aabbccdd&quot;, k = 7</span></p>

<p><strong>Output:</strong> <span class="example-io">5</span></p>

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

<p>The possible strings are: <code>&quot;aabbccdd&quot;</code>, <code>&quot;aabbccd&quot;</code>, <code>&quot;aabbcdd&quot;</code>, <code>&quot;aabccdd&quot;</code>, and <code>&quot;abbccdd&quot;</code>.</p>
</div>

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

<div class="example-block">
<p><strong>Input:</strong> <span class="example-io">word = &quot;aabbccdd&quot;, k = 8</span></p>

<p><strong>Output:</strong> <span class="example-io">1</span></p>

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

<p>The only possible string is <code>&quot;aabbccdd&quot;</code>.</p>
</div>

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

<div class="example-block">
<p><strong>Input:</strong> <span class="example-io">word = &quot;aaabbb&quot;, k = 3</span></p>

<p><strong>Output:</strong> <span class="example-io">8</span></p>
</div>

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

<ul>
	<li><code>1 &lt;= word.length &lt;= 5 * 10<sup>5</sup></code></li>
	<li><code>word</code> consists only of lowercase English letters.</li>
	<li><code>1 &lt;= k &lt;= 2000</code></li>
</ul>
String; Dynamic Programming; Prefix Sum
function possibleStringCount(word: string, k: number): number {
    const mod = 1_000_000_007;
    const nums: number[] = [];
    let ans = 1;
    let cur = 0;
    const n = word.length;

    for (let i = 0; i < n; i++) {
        cur++;
        if (i === n - 1 || word[i] !== word[i + 1]) {
            if (cur > 1) {
                if (k > 0) {
                    nums.push(cur - 1);
                }
                ans = (ans * cur) % mod;
            }
            cur = 0;
            k--;
        }
    }

    if (k < 1) {
        return ans;
    }

    const m = nums.length;
    const f: number[][] = Array.from({ length: m + 1 }, () => Array(k).fill(0));
    f[0][0] = 1;

    for (let i = 1; i <= m; i++) {
        const x = nums[i - 1];
        const s: number[] = Array(k + 1).fill(0);
        for (let j = 0; j < k; j++) {
            s[j + 1] = (s[j] + f[i - 1][j]) % mod;
        }
        for (let j = 0; j < k; j++) {
            const l = Math.max(0, j - x);
            f[i][j] = (s[j + 1] - s[l] + mod) % mod;
        }
    }

    let sum = 0;
    for (let j = 0; j < k; j++) {
        sum = (sum + f[m][j]) % mod;
    }

    return (ans - sum + mod) % mod;
}
