
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 \( a, b, c \) be the sides opposite to the interior angles \( A, B, C \) of triangle \( \triangle ABC \), and let the area of the triangle be \( S = \frac{1}{2} c^2 \). If \( ab = \sqrt{2} \), then the maximum value of \( a^2 + b^2 + c^2 \) is .

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