<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]:
        digits = [int(dig) for dig in num]
        operations = ('+', '-')
        op_to_func = {'+': lambda x, y: x + y,
                      '-': lambda x, y: x - y}
        so_far = ['+']
        works = []
        def inner(index: int, res: int, prod: int, prod_pre_conc: int, conc: int, op: str):
            if so_far == ['+', '+', '']:
                print(res)
            if index == len(num):
                if res == target:
                    works.append(so_far[:])
            else:
                digit = digits[index]

                so_far.append('*')
                inner(index + 1, op_to_func[op](res, prod * (digits[index] - 1)), prod * digit, prod, digit, op)
                so_far.pop()

                for operation in operations:
                    so_far.append(operation)
                    inner(index + 1, op_to_func[operation](res, digit), digit, 1, digit, operation)
                    so_far.pop()

                if conc != 0:
                    new_conc = conc * 10 + digit
                    new_prod = prod_pre_conc * new_conc
                    so_far.append('')
                    if res == 3:
                        print(prod, new_prod)
                    inner(index + 1, op_to_func[op](res, new_prod - prod), new_prod, prod_pre_conc, new_conc, op)
                    so_far.pop()



        inner(1, digits[0], digits[0], 1, digits[0], '+')

        possibilities = []
        for sequence in works:
            temp = [num[0]]
            for i in range(1, len(num)):
                temp.append(sequence[i])
                temp.append(num[i])
            possibilities.append(''.join(temp))
        return possibilities

        
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