
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
In triangle \( \triangle ABC \), the interior angles \( \angle A, \angle B, \angle C \) have opposite sides with lengths \( a, b, c \) respectively. Given that \( c = 1 \) and the area of \( \triangle ABC \) is equal to \( \frac{a^{2} + b^{2} - 1}{4} \), find the maximum value of the area of \( \triangle ABC \).

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