
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
Given that \( f(x) \) is an even function defined on \(\mathbf{R}\) and that \( f(x+1) = f(x-1) \). For \( 1 \leqslant x \leqslant 2 \), \( f(x) = \log_{a} x \) where \( a > 0 \) and \( a \neq 1 \). Determine the analytic expression of \( f(x) \) for \( -1 \leqslant x \leqslant 1 \).

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