
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
The function \( y = f(x) \) is an even function and a periodic function with a period of 2. When \( x \in [2, 3] \), \( f(x) = x - 1 \). On the graph of \( y = f(x) \), there are two points \( A \) and \( B \) such that their y-coordinates are equal (point \( A \) is to the left of point \( B \)), and their x-coordinates lie in the interval \( [1, 3] \). The fixed point \( C \) has coordinates \( (0, a) \), where \( a > 2 \). Find the maximum area of triangle \( \triangle ABC \) in terms of \( a \).

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