
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
Let  $x \neq y$  be positive reals satisfying  $x^3+2013y=y^3+2013x$ , and let  $M = \left( \sqrt{3}+1 \right)x + 2y$ .  Determine the maximum possible value of  $M^2$ .

*Proposed by Varun Mohan*

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