
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
If the function $y=f(x)$ is an even function with a period of 2 and when $x \in [2, 3]$, $f(x) = x-1$.
(1) Find the explicit expression for $f(x)$ when $x \in [1, 2]$;
(2) On the graph of $y=f(x)$, there are two points A and B with equal ordinates, and their abscissas both lie in the interval $[1, 3]$. A fixed point C is given by coordinates $(0, a)$ (where $2 < a < 3$), find the maximum value of the area of triangle $ABC$.

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