<p>A sequence is <strong>special</strong> if it consists of a <strong>positive</strong> number of <code>0</code>s, followed by a <strong>positive</strong> number of <code>1</code>s, then a <strong>positive</strong> number of <code>2</code>s.</p>

<ul>
	<li>For example, <code>[0,1,2]</code> and <code>[0,0,1,1,1,2]</code> are special.</li>
	<li>In contrast, <code>[2,1,0]</code>, <code>[1]</code>, and <code>[0,1,2,0]</code> are not special.</li>
</ul>

<p>Given an array <code>nums</code> (consisting of <strong>only</strong> integers <code>0</code>, <code>1</code>, and <code>2</code>), return<em> the <strong>number of different subsequences</strong> that are special</em>. Since the answer may be very large, <strong>return it modulo </strong><code>10<sup>9</sup> + 7</code>.</p>

<p>A <strong>subsequence</strong> of an array is a sequence that can be derived from the array by deleting some or no elements without changing the order of the remaining elements. Two subsequences are <strong>different</strong> if the <strong>set of indices</strong> chosen are different.</p>

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

<pre>
<strong>Input:</strong> nums = [0,1,2,2]
<strong>Output:</strong> 3
<strong>Explanation:</strong> The special subsequences are bolded [<strong><u>0</u></strong>,<strong><u>1</u></strong>,<strong><u>2</u></strong>,2], [<strong><u>0</u></strong>,<strong><u>1</u></strong>,2,<strong><u>2</u></strong>], and [<strong><u>0</u></strong>,<strong><u>1</u></strong>,<strong><u>2</u></strong>,<strong><u>2</u></strong>].
</pre>

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

<pre>
<strong>Input:</strong> nums = [2,2,0,0]
<strong>Output:</strong> 0
<strong>Explanation:</strong> There are no special subsequences in [2,2,0,0].
</pre>

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

<pre>
<strong>Input:</strong> nums = [0,1,2,0,1,2]
<strong>Output:</strong> 7
<strong>Explanation:</strong> The special subsequences are bolded:
- [<strong><u>0</u></strong>,<strong><u>1</u></strong>,<strong><u>2</u></strong>,0,1,2]
- [<strong><u>0</u></strong>,<strong><u>1</u></strong>,2,0,1,<strong><u>2</u></strong>]
- [<strong><u>0</u></strong>,<strong><u>1</u></strong>,<strong><u>2</u></strong>,0,1,<strong><u>2</u></strong>]
- [<strong><u>0</u></strong>,<strong><u>1</u></strong>,2,0,<strong><u>1</u></strong>,<strong><u>2</u></strong>]
- [<strong><u>0</u></strong>,1,2,<strong><u>0</u></strong>,<strong><u>1</u></strong>,<strong><u>2</u></strong>]
- [<strong><u>0</u></strong>,1,2,0,<strong><u>1</u></strong>,<strong><u>2</u></strong>]
- [0,1,2,<strong><u>0</u></strong>,<strong><u>1</u></strong>,<strong><u>2</u></strong>]
</pre>

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

<ul>
	<li><code>1 &lt;= nums.length &lt;= 10<sup>5</sup></code></li>
	<li><code>0 &lt;= nums[i] &lt;= 2</code></li>
</ul>

class Solution:
    def countSpecialSubsequences(self, nums: List[int]) -> int:
        n = len(nums)
        mod = 10**9 + 7

        memo = [[-1] * 4 for _ in range(n)]
        
        def rec(i, prev):
            if i >= n:
                return int(prev == 2)

            if prev > 2:
                return 0

            if memo[i][prev] != -1:
                return memo[i][prev]

            if prev == -1:
                if nums[i] == 0:
                    memo[i][prev] = rec(i + 1, 0) + rec(i + 1, prev)
                else:
                    memo[i][prev] = rec(i + 1, prev)
                return memo[i][prev]

            if nums[i] == prev:
                memo[i][prev] = (2 * rec(i + 1, prev)) % mod
                return memo[i][prev]
            elif nums[i] == prev + 1:
                memo[i][prev] = (rec(i + 1, nums[i]) + rec(i + 1, prev)) % mod
                return memo[i][prev]
            else:
                memo[i][prev] = rec(i + 1, prev)
                return memo[i][prev]

        return rec(0, -1) % mod
57400