<p>You have a pointer at index <code>0</code> in an array of size <code>arrLen</code>. At each step, you can move 1 position to the left, 1 position to the right in the array, or stay in the same place (The pointer should not be placed outside the array at any time).</p>

<p>Given two integers <code>steps</code> and <code>arrLen</code>, return the number of ways such that your pointer is still at index <code>0</code> after <strong>exactly</strong> <code>steps</code> steps. Since the answer may be too 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>

<pre>
<strong>Input:</strong> steps = 3, arrLen = 2
<strong>Output:</strong> 4
<strong>Explanation: </strong>There are 4 differents ways to stay at index 0 after 3 steps.
Right, Left, Stay
Stay, Right, Left
Right, Stay, Left
Stay, Stay, Stay
</pre>

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

<pre>
<strong>Input:</strong> steps = 2, arrLen = 4
<strong>Output:</strong> 2
<strong>Explanation:</strong> There are 2 differents ways to stay at index 0 after 2 steps
Right, Left
Stay, Stay
</pre>

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

<pre>
<strong>Input:</strong> steps = 4, arrLen = 2
<strong>Output:</strong> 8
</pre>

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

<ul>
	<li><code>1 &lt;= steps &lt;= 500</code></li>
	<li><code>1 &lt;= arrLen &lt;= 10<sup>6</sup></code></li>
</ul>

class Solution:
    def _mod(self, num: int) -> int:
        return num % (pow(10, 9) + 7)
    
    def numWays(self, steps: int, arrLen: int) -> int:
        memo = {}
        def top_down(cur_pos: int, target_pos: int, steps: int) -> int:
            nonlocal arrLen
            
            key = (cur_pos, target_pos, steps)
            
            if key not in memo:
                # If not enough steps to reach
                if abs(cur_pos - target_pos) > steps:
                    return -1

                # If no steps left, check if we at target
                if steps == 0:
                    return 1 if cur_pos == target_pos else -1

                # Explore the 3 options
                # Choose to stay
                ways_stay = max(0, top_down(cur_pos, target_pos, steps-1))

                # Choose to go left
                ways_left = 0
                if cur_pos - 1 >= 0:
                    ways_left = max(0, top_down(cur_pos - 1, target_pos, steps-1))

                # Choose to go right
                ways_right = 0
                if cur_pos + 1 < arrLen:
                    ways_right = max(0, top_down(cur_pos + 1, target_pos, steps-1))

                total_ways = ways_stay + ways_left + ways_right
                memo[key] = self._mod(total_ways)
                
            return memo[key]
        return top_down(0, 0, steps)
54198