<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]:
        answer = set()
        memo = {}
    
        def dp(idx, total, path, last_number):
            if idx == len(num):
                if total == target:
                    answer.add(path)
                return

            state = (idx, total, path, last_number)
            if state in memo:
                return
            memo[state] = True
        
            for i in range(idx, len(num)):
                if len(num[idx:i+1]) > 1 and num[idx:i+1][0] == "0":
                    continue
                
                tmp_number = num[idx:i+1]
            
                if last_number == "":
                    dp(i + 1, int(tmp_number), tmp_number, tmp_number)
                else:

                    dp(i + 1, total + int(tmp_number), path + "+" + tmp_number, tmp_number)

                    dp(i + 1, total - int(tmp_number), path + "-" + tmp_number, "-" + tmp_number)

                    dp(i + 1, total - int(last_number) + (int(last_number) * int(tmp_number)), path + "*" +         tmp_number, str(int(tmp_number) * int(last_number)))

        dp(0, 0, "", "")
        return list(answer)
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