
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
Find integers $a$, $b$, and $c$ in the simplest form such that $\sqrt{11} + \frac{2}{\sqrt{11}} + \sqrt{2} + \frac{3}{\sqrt{2}}$ can be expressed as $\frac{a\sqrt{11} + b\sqrt{2}}{c}$, where $c$ is minimized.

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