<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]:
        dp = [num[0]]
        for i in range(1, len(num)):
            curr = []
            for entry in dp[-1]:
                curr.append(entry+''+num[i])
                curr.append(entry+'+'+num[i])
                curr.append(entry+'-'+num[i])
                curr.append(entry+'*'+num[i])
            dp.append(curr)
            
        def evaluate(x):
            if '*' in x:
                i = 0
                while x[i] != '*': i+= 1
                op = i
                i = op+1
                while i < len(x) and \
                      x[i] != '+' and \
                      x[i] != '-' and \
                      x[i] != '*': i+= 1
                end = i
                i = op-1
                while i > -1 and \
                      x[i] != '+' and \
                      x[i] != '-' and \
                      x[i] != '*': i-= 1
                start = i
                num1 = x[start+1:op]
                num2 = x[op+1:end]
                return evaluate(x[:start+1]+str(int(num1)*int(num2))+x[end:])
            elif '+' not in x and '-' not in x: return int(x)
            elif x[0] == '-' and '+' not in x[1:] and '-' not in x[1:]: return int(x)
            else:
                i = 1
                while x[i] != '+' and x[i] != '-': i+= 1
                op = i
                i += 1
                while i < len(x) and \
                    x[i] != '+' and x[i] != '-': i+= 1
                end = i
                num1 = x[:op]
                num2 = x[op+1:end]
                if x[op] == '+':
                    return evaluate(str(int(num1)+int(num2))+x[end:])
                else: return evaluate(str(int(num1)-int(num2))+x[end:])

        def validate(x):
            if x[0] == '0' and x[1] not in '+-*': return False
            for i in range(len(x)-2):
                if x[i] in '+-*' and \
                   x[i+1] == '0' and \
                   x[i+2] not in '+-*': return False
            return True
            
        ret = []
        for candidate in dp[-1]:
            if not validate(candidate): continue
            if evaluate(candidate) == target: ret.append(candidate)
        return ret
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