<p>We had some 2-dimensional coordinates, like <code>&quot;(1, 3)&quot;</code> or <code>&quot;(2, 0.5)&quot;</code>. Then, we removed all commas, decimal points, and spaces and ended up with the string s.</p>

<ul>
	<li>For example, <code>&quot;(1, 3)&quot;</code> becomes <code>s = &quot;(13)&quot;</code> and <code>&quot;(2, 0.5)&quot;</code> becomes <code>s = &quot;(205)&quot;</code>.</li>
</ul>

<p>Return <em>a list of strings representing all possibilities for what our original coordinates could have been</em>.</p>

<p>Our original representation never had extraneous zeroes, so we never started with numbers like <code>&quot;00&quot;</code>, <code>&quot;0.0&quot;</code>, <code>&quot;0.00&quot;</code>, <code>&quot;1.0&quot;</code>, <code>&quot;001&quot;</code>, <code>&quot;00.01&quot;</code>, or any other number that can be represented with fewer digits. Also, a decimal point within a number never occurs without at least one digit occurring before it, so we never started with numbers like <code>&quot;.1&quot;</code>.</p>

<p>The final answer list can be returned in any order. All coordinates in the final answer have exactly one space between them (occurring after the comma.)</p>

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

<pre>
<strong>Input:</strong> s = &quot;(123)&quot;
<strong>Output:</strong> [&quot;(1, 2.3)&quot;,&quot;(1, 23)&quot;,&quot;(1.2, 3)&quot;,&quot;(12, 3)&quot;]
</pre>

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

<pre>
<strong>Input:</strong> s = &quot;(0123)&quot;
<strong>Output:</strong> [&quot;(0, 1.23)&quot;,&quot;(0, 12.3)&quot;,&quot;(0, 123)&quot;,&quot;(0.1, 2.3)&quot;,&quot;(0.1, 23)&quot;,&quot;(0.12, 3)&quot;]
<strong>Explanation:</strong> 0.0, 00, 0001 or 00.01 are not allowed.
</pre>

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

<pre>
<strong>Input:</strong> s = &quot;(00011)&quot;
<strong>Output:</strong> [&quot;(0, 0.011)&quot;,&quot;(0.001, 1)&quot;]
</pre>

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

<ul>
	<li><code>4 &lt;= s.length &lt;= 12</code></li>
	<li><code>s[0] == &#39;(&#39;</code> and <code>s[s.length - 1] == &#39;)&#39;</code>.</li>
	<li>The rest of <code>s</code> are digits.</li>
</ul>

class Solution:
    def ambiguousCoordinates(self, s: str) -> List[str]:
        def is_valid_number(l):
            if l[0] == "0" and len(l) > 1 and "." not in l:
                return False
            if "." in l:
                front, back = ''.join(l).split(".")
                if all([b == "0" for b in back]) or back[-1] == "0":
                    return False
            return True

        def is_valid_pre_decimal(l):
            if (is_valid_number(l) or (l[0] == "0" and len(l) == 1)) and "." not in l:
                return True
            return False            

        def generate_numbers(s_idx, working_number, curr_number):
            if s_idx == len(s):
                if working_number and curr_number and is_valid_number(curr_number):
                    return [f"({working_number}, {''.join(curr_number)})"]
                return []

            include_in_curr_number = generate_numbers(s_idx + 1, working_number, curr_number + [s[s_idx]])

            add_decimal = []
            if curr_number and is_valid_pre_decimal(curr_number):
                add_decimal = generate_numbers(s_idx + 1, working_number, curr_number + [".", s[s_idx]])

            create_new_number = []
            if not working_number and curr_number and is_valid_number(curr_number):
                create_new_number = generate_numbers(s_idx + 1, ''.join(curr_number), [s[s_idx]])

            return include_in_curr_number + create_new_number + add_decimal

        s = s[1:-1]

        return generate_numbers(0, None, [])
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