<p>We have two arrays <code>arr1</code> and <code>arr2</code> which are initially empty. You need to add positive integers to them such that they satisfy all the following conditions:</p>

<ul>
	<li><code>arr1</code> contains <code>uniqueCnt1</code> <strong>distinct</strong> positive integers, each of which is <strong>not divisible</strong> by <code>divisor1</code>.</li>
	<li><code>arr2</code> contains <code>uniqueCnt2</code> <strong>distinct</strong> positive integers, each of which is <strong>not divisible</strong> by <code>divisor2</code>.</li>
	<li><strong>No</strong> integer is present in both <code>arr1</code> and <code>arr2</code>.</li>
</ul>

<p>Given <code>divisor1</code>, <code>divisor2</code>, <code>uniqueCnt1</code>, and <code>uniqueCnt2</code>, return <em>the <strong>minimum possible maximum</strong> integer that can be present in either array</em>.</p>

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

<pre>
<strong>Input:</strong> divisor1 = 2, divisor2 = 7, uniqueCnt1 = 1, uniqueCnt2 = 3
<strong>Output:</strong> 4
<strong>Explanation:</strong> 
We can distribute the first 4 natural numbers into arr1 and arr2.
arr1 = [1] and arr2 = [2,3,4].
We can see that both arrays satisfy all the conditions.
Since the maximum value is 4, we return it.
</pre>

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

<pre>
<strong>Input:</strong> divisor1 = 3, divisor2 = 5, uniqueCnt1 = 2, uniqueCnt2 = 1
<strong>Output:</strong> 3
<strong>Explanation:</strong> 
Here arr1 = [1,2], and arr2 = [3] satisfy all conditions.
Since the maximum value is 3, we return it.</pre>

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

<pre>
<strong>Input:</strong> divisor1 = 2, divisor2 = 4, uniqueCnt1 = 8, uniqueCnt2 = 2
<strong>Output:</strong> 15
<strong>Explanation:</strong> 
Here, the final possible arrays can be arr1 = [1,3,5,7,9,11,13,15], and arr2 = [2,6].
It can be shown that it is not possible to obtain a lower maximum satisfying all conditions. 
</pre>

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

<ul>
	<li><code>2 &lt;= divisor1, divisor2 &lt;= 10<sup>5</sup></code></li>
	<li><code>1 &lt;= uniqueCnt1, uniqueCnt2 &lt; 10<sup>9</sup></code></li>
	<li><code>2 &lt;= uniqueCnt1 + uniqueCnt2 &lt;= 10<sup>9</sup></code></li>
</ul>

from math import gcd

class Solution:
    # Function to compute LCM of two numbers
    def calculateLCM(self, a, b):
        return (a * b) // gcd(a, b)
    
    # Helper function to determine if a maximum value is valid
    def isValid(self, maxValue, divisor1, divisor2, uniqueCnt1, uniqueCnt2):
        lcm = self.calculateLCM(divisor1, divisor2)
        
        # Count of numbers not divisible by divisor1
        count1 = maxValue - (maxValue // divisor1)
        # Count of numbers not divisible by divisor2
        count2 = maxValue - (maxValue // divisor2)
        # Count of numbers not divisible by both divisors
        countBoth = maxValue - (maxValue // lcm)
        
        # Check if we can pick uniqueCnt1 numbers for arr1 and uniqueCnt2 for arr2
        return (count1 >= uniqueCnt1) and (count2 >= uniqueCnt2) and (countBoth >= (uniqueCnt1 + uniqueCnt2))
    
    # Main function to find the minimum possible maximum integer
    def minimizeSet(self, divisor1, divisor2, uniqueCnt1, uniqueCnt2):
        left, right = 1, 10**18  # Define the search range for maxValue
        
        # Binary search to find the minimum valid maxValue
        while left < right:
            mid = (left + right) // 2
            
            if self.isValid(mid, divisor1, divisor2, uniqueCnt1, uniqueCnt2):
                right = mid  # If valid, search the left half
            else:
                left = mid + 1  # If not valid, search the right half
        
        return left
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