<p>You are given a string <code>s</code> and two integers <code>x</code> and <code>y</code>. You can perform two types of operations any number of times.</p>

<ul>
	<li>Remove substring <code>&quot;ab&quot;</code> and gain <code>x</code> points.

	<ul>
		<li>For example, when removing <code>&quot;ab&quot;</code> from <code>&quot;c<u>ab</u>xbae&quot;</code> it becomes <code>&quot;cxbae&quot;</code>.</li>
	</ul>
	</li>
	<li>Remove substring <code>&quot;ba&quot;</code> and gain <code>y</code> points.
	<ul>
		<li>For example, when removing <code>&quot;ba&quot;</code> from <code>&quot;cabx<u>ba</u>e&quot;</code> it becomes <code>&quot;cabxe&quot;</code>.</li>
	</ul>
	</li>
</ul>

<p>Return <em>the maximum points you can gain after applying the above operations on</em> <code>s</code>.</p>

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

<pre>
<strong>Input:</strong> s = &quot;cdbcbbaaabab&quot;, x = 4, y = 5
<strong>Output:</strong> 19
<strong>Explanation:</strong>
- Remove the &quot;ba&quot; underlined in &quot;cdbcbbaaa<u>ba</u>b&quot;. Now, s = &quot;cdbcbbaaab&quot; and 5 points are added to the score.
- Remove the &quot;ab&quot; underlined in &quot;cdbcbbaa<u>ab</u>&quot;. Now, s = &quot;cdbcbbaa&quot; and 4 points are added to the score.
- Remove the &quot;ba&quot; underlined in &quot;cdbcb<u>ba</u>a&quot;. Now, s = &quot;cdbcba&quot; and 5 points are added to the score.
- Remove the &quot;ba&quot; underlined in &quot;cdbc<u>ba</u>&quot;. Now, s = &quot;cdbc&quot; and 5 points are added to the score.
Total score = 5 + 4 + 5 + 5 = 19.</pre>

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

<pre>
<strong>Input:</strong> s = &quot;aabbaaxybbaabb&quot;, x = 5, y = 4
<strong>Output:</strong> 20
</pre>

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

<ul>
	<li><code>1 &lt;= s.length &lt;= 10<sup>5</sup></code></li>
	<li><code>1 &lt;= x, y &lt;= 10<sup>4</sup></code></li>
	<li><code>s</code> consists of lowercase English letters.</li>
</ul>

class Solution:
    def maximumGain(self, s: str, x: int, y: int) -> int:
        ab_strs = self.find_ab_strings(s)

        gain = 0
        for ab_str in ab_strs:
            score = self.find_max_score(ab_str, x, y)
            gain += score

        return gain

    def find_ab_strings(self, s):
        ab_strs = []
        ab_str = []
        for i in range(len(s)):
            c = s[i]
            if c == "a" or c == "b":
                ab_str.append(c)

                if i + 1 == len(s) or (s[i + 1] != "a" and s[i + 1] != "b"):
                    ab_strs.append("".join(ab_str))
                    ab_str = []

        return ab_strs

    def find_max_score(self, s, x, y):
        n = len(s)
        count_a = 0
        count_b = 0
        for c in s:
            if c == "a":
                count_a += 1

        count_b = n - count_a
        pair_count = min(count_a, count_b)

        count_a_remaining = 0
        count_ab = 0
        for c in s:
            if c == "a":
                count_a_remaining += 1
            else:
                if count_a_remaining > 0:
                    count_ab += 1
                    count_a_remaining -= 1

        score1 = count_ab * x + (pair_count - count_ab) * y
        
        count_b_remaining = 0
        count_ba = 0
        for c in s:
            if c == "b":
                count_b_remaining += 1
            else:
                if count_b_remaining > 0:
                    count_ba += 1
                    count_b_remaining -= 1

        score2 = (pair_count - count_ba) * x + count_ba * y
        
        return max(score1, score2)

        
        
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