
When tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.

[ASSESS]

[ADVANCE]

[VERIFY]

[SIMPLIFY]

[SYNTHESIZE]

[PIVOT]

[OUTPUT]

You should strictly follow the format below:

[ACTION NAME]

# Your action step 1

# Your action step 2

# Your action step 3

...

Next action: [NEXT ACTION NAME]


system
Given two positive integers \(x\) and \(y\), \(xy - (x + y) = \operatorname{HCF}(x, y) + \operatorname{LCM}(x, y)\), where \(\operatorname{HCF}(x, y)\) and \(\operatorname{LCM}(x, y)\) are respectively the greatest common divisor and the least common multiple of \(x\) and \(y\). If \(c\) is the maximum possible value of \(x + y\), find \(c\).

Present the answer in LaTex format: \boxed{Your answer}
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