<p>You are given a <strong>0-indexed</strong> string <code>s</code>, a string <code>a</code>, a string <code>b</code>, and an integer <code>k</code>.</p>

<p>An index <code>i</code> is <strong>beautiful</strong> if:</p>

<ul>
	<li><code>0 &lt;= i &lt;= s.length - a.length</code></li>
	<li><code>s[i..(i + a.length - 1)] == a</code></li>
	<li>There exists an index <code>j</code> such that:
	<ul>
		<li><code>0 &lt;= j &lt;= s.length - b.length</code></li>
		<li><code>s[j..(j + b.length - 1)] == b</code></li>
		<li><code>|j - i| &lt;= k</code></li>
	</ul>
	</li>
</ul>

<p>Return <em>the array that contains beautiful indices in <strong>sorted order from smallest to largest</strong></em>.</p>

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

<pre>
<strong>Input:</strong> s = &quot;isawsquirrelnearmysquirrelhouseohmy&quot;, a = &quot;my&quot;, b = &quot;squirrel&quot;, k = 15
<strong>Output:</strong> [16,33]
<strong>Explanation:</strong> There are 2 beautiful indices: [16,33].
- The index 16 is beautiful as s[16..17] == &quot;my&quot; and there exists an index 4 with s[4..11] == &quot;squirrel&quot; and |16 - 4| &lt;= 15.
- The index 33 is beautiful as s[33..34] == &quot;my&quot; and there exists an index 18 with s[18..25] == &quot;squirrel&quot; and |33 - 18| &lt;= 15.
Thus we return [16,33] as the result.
</pre>

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

<pre>
<strong>Input:</strong> s = &quot;abcd&quot;, a = &quot;a&quot;, b = &quot;a&quot;, k = 4
<strong>Output:</strong> [0]
<strong>Explanation:</strong> There is 1 beautiful index: [0].
- The index 0 is beautiful as s[0..0] == &quot;a&quot; and there exists an index 0 with s[0..0] == &quot;a&quot; and |0 - 0| &lt;= 4.
Thus we return [0] as the result.
</pre>

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

<ul>
	<li><code>1 &lt;= k &lt;= s.length &lt;= 5 * 10<sup>5</sup></code></li>
	<li><code>1 &lt;= a.length, b.length &lt;= 5 * 10<sup>5</sup></code></li>
	<li><code>s</code>, <code>a</code>, and <code>b</code> contain only lowercase English letters.</li>
</ul>

class Solution:
    def beautifulIndices(self, s: str, a: str, b: str, k: int) -> List[int]:
                # z-function
        # prefix = abc s = "aabcaa"
        #                   130011
        # go through, find all matches, and find the closest index in the a_indices array that matches
        # if there is a match, add it to the result
        def z_function(target, s):
            prefixed = target + s
            z = [0] * len(prefixed)
            left = right = 0
            for i in range(1, len(prefixed)):
                if i < right:
                    z[i] = min(right - i, z[i-left])
                while i + z[i] < len(prefixed) and prefixed[i + z[i]] == prefixed[z[i]]:
                    z[i] += 1
                if i + z[i] > right:
                    left = i
                    right = i + z[i]
            return z[len(target):]
        z_a = z_function(a, s)
        z_b = z_function(b, s)
        a_indices = [i for i in range(len(s)) if z_a[i] >= len(a)]
        b_indices = [i for i in range(len(s)) if z_b[i] >= len(b)]
        # print(z_a)
        # print(z_b)
        # print(a_indices)
        # print(b_indices)
        # [5,7,10,15]
        # 12
        def has_b_within_k(x):
            if not b_indices:
                return False
            idx = bisect_left(b_indices, x)
            if idx > 0:
                idx -= 1
            return abs(b_indices[idx] - x) <= k or (idx+1 < len(b_indices) and abs(b_indices[idx+1] - x) <= k)
        return [x for x in a_indices if has_b_within_k(x)]
        
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