<p>Given a string <code>s</code>, return <em>the number of <strong>palindromic substrings</strong> in it</em>.</p>

<p>A string is a <strong>palindrome</strong> when it reads the same backward as forward.</p>

<p>A <strong>substring</strong> is a contiguous sequence of characters within the string.</p>

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

<pre>
<strong>Input:</strong> s = &quot;abc&quot;
<strong>Output:</strong> 3
<strong>Explanation:</strong> Three palindromic strings: &quot;a&quot;, &quot;b&quot;, &quot;c&quot;.
</pre>

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

<pre>
<strong>Input:</strong> s = &quot;aaa&quot;
<strong>Output:</strong> 6
<strong>Explanation:</strong> Six palindromic strings: &quot;a&quot;, &quot;a&quot;, &quot;a&quot;, &quot;aa&quot;, &quot;aa&quot;, &quot;aaa&quot;.
</pre>

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

<ul>
	<li><code>1 &lt;= s.length &lt;= 1000</code></li>
	<li><code>s</code> consists of lowercase English letters.</li>
</ul>

class Solution {
public:
    int countSubstrings(string s) {
        int n = s.size();
        vector<vector<pair<bool, int>>> dp(n, vector<pair<bool, int>>(n, {false, 0}));
        for (int i = 0; i < n; i++) {
            dp[i][i] = {true, 1};
        }
        for (int j = 1; j < n; j++) {
            for (int i = j - 1; i >= 0; i--) {
                if (s[i] == s[j]) {
                    if (i + 1 <= j - 1) {
                        dp[i][j].first = dp[i + 1][j - 1].first;
                    } else {
                        dp[i][j].first = true;
                    }
                }
                dp[i][j].second = dp[i + 1][j].second + dp[i][j - 1].second - dp[i + 1][j - 1].second;
                if (dp[i][j].first) {
                    dp[i][j].second++;
                }
            }
        }
        return dp[0][n - 1].second;
    }
};

// Example 1:
// Input: s = "abc"
// Output: 3
// Explanation: Three palindromic strings: "a", "b", "c".

// Example 2:
// Input: s = "aaa"
// Output: 6
// Explanation: Six palindromic strings: "a", "a", "a", "aa", "aa", "aaa".

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