<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>

from typing import List

class Solution:
    def __init__(self):
        self.known_ao = {}
        self.known_mul = {}

    def check_multiplication(self, num, target):
        if len(num) < len(str(target)):
            return []

        if len(num) == 1:
            return [num] if int(num) == target else []

        if num[0] == '0':
            if target == 0:
                return ['0*' + option for option, amount in self.all_multiplication(num[1:])]
            else:
                return []

        options = []

        if int(num) == target:
            options.append(num)

        for i in range(1, len(num)):
            p = int(num[:i])
            if not target % p:
                for option in self.check_multiplication(num[i:], target // p):
                    options.append(num[:i] + '*' + option)

        return options

    def all_multiplication(self, num):
        if num in self.known_mul:
            return self.known_mul[num]

        options = [(num, int(num))] if num[0] != '0' or len(num) == 1 else []
        until = 2 if num[0] == '0' and len(num) > 1 else len(num)

        for i in range(1, until):
            p = int(num[:i])

            for option, amount in self.all_multiplication(num[i:]):
                options.append((num[:i] + '*' + option, amount * p))

        self.known_mul[num] = options
        return self.known_mul[num]

    def addOperators(self, num: str, target: int) -> List[str]:
        if (num, target) in self.known_ao:
            return self.known_ao[(num, target)]

        if len(num) < len(str(target)):  # Not enough digits to reach target
            return []

        if len(num) == 1:
            return [num] if int(num) == target else []

        options = []
        options.extend(self.check_multiplication(num, target))

        for i in range(1, len(num)):
            for option, amount in self.all_multiplication(num[:i]):
                for option2 in self.addOperators(num[i:], target - amount):
                    options.append(option + '+' + option2)

                for option2 in self.addOperators(num[i:], amount - target):
                    options.append(option + '-' + option2.replace('-', 'X').replace('+', '-').replace('X', '+'))

        self.known_ao[(num, target)] = options
        return self.known_ao[(num, target)]

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