<p>Given a string <code>num</code> that contains only digits and an integer <code>target</code>, return <em><strong>all possibilities</strong> to insert the binary operators </em><code>&#39;+&#39;</code><em>, </em><code>&#39;-&#39;</code><em>, and/or </em><code>&#39;*&#39;</code><em> between the digits of </em><code>num</code><em> so that the resultant expression evaluates to the </em><code>target</code><em> value</em>.</p>

<p>Note that operands in the returned expressions <strong>should not</strong> contain leading zeros.</p>

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

<pre>
<strong>Input:</strong> num = &quot;123&quot;, target = 6
<strong>Output:</strong> [&quot;1*2*3&quot;,&quot;1+2+3&quot;]
<strong>Explanation:</strong> Both &quot;1*2*3&quot; and &quot;1+2+3&quot; evaluate to 6.
</pre>

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

<pre>
<strong>Input:</strong> num = &quot;232&quot;, target = 8
<strong>Output:</strong> [&quot;2*3+2&quot;,&quot;2+3*2&quot;]
<strong>Explanation:</strong> Both &quot;2*3+2&quot; and &quot;2+3*2&quot; evaluate to 8.
</pre>

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

<pre>
<strong>Input:</strong> num = &quot;3456237490&quot;, target = 9191
<strong>Output:</strong> []
<strong>Explanation:</strong> There are no expressions that can be created from &quot;3456237490&quot; to evaluate to 9191.
</pre>

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

<ul>
	<li><code>1 &lt;= num.length &lt;= 10</code></li>
	<li><code>num</code> consists of only digits.</li>
	<li><code>-2<sup>31</sup> &lt;= target &lt;= 2<sup>31</sup> - 1</code></li>
</ul>

class Solution:
    def addOperators(self, num: str, target: int) -> List[str]:
        result = []

        def backtrack(i, prev_val, prev_total, curr):
            if i == len(num):
                if prev_total == target:
                    result.append(''.join(curr))
                return

            curr_val = 0
            for mid in range(i, len(num)):
                curr_val = curr_val * 10 + int(num[mid])
                if i == 0:
                    backtrack(mid + 1, curr_val, curr_val, [num[:mid + 1]])
                else:
                    curr.append('+' + num[i:mid + 1])
                    backtrack(mid + 1, curr_val, prev_total + curr_val, curr)
                    curr.pop()
                    curr.append('-' + num[i:mid + 1])
                    backtrack(mid + 1, - curr_val, prev_total - curr_val, curr)
                    curr.pop()
                    curr.append('*' + num[i:mid + 1])
                    backtrack(mid + 1, prev_val * curr_val, prev_total - prev_val + prev_val * curr_val, curr)
                    curr.pop()
                if num[i] == '0':
                    break
    

        backtrack(0, 0, 0, [])
        return result




    ''' backtrack, full dfs, very slow... '''
    # def addOperators(self, num: str, target: int) -> List[str]:
    #     result = []

    #     def backtrack(curr, i):
    #         if i == len(num):
    #             string = ''.join(curr)
    #             if eval(string) == target:
    #                 result.append(string)
    #             return

    #         if num[i] == '0':
    #             for sign in ('+', '-', '*'):
    #                 curr.append(sign + num[i])
    #                 backtrack(curr, i + 1)
    #                 curr.pop()
    #             return

    #         for j in range(i + 1, len(num) + 1):
    #             for sign in ('+', '-', '*'):
    #                 curr.append(sign + num[i:j])
    #                 backtrack(curr, j)
    #                 curr.pop()

    #     if num[0] == '0':
    #         backtrack([num[0]], 1)
    #     else:
    #         for i in range(1, len(num) + 1):
    #             backtrack([num[:i]], i)
    #     return result

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