
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 \( f(x) \) be an even function defined on \(\mathbf{R}\) with a period of 2 and strictly decreasing on the interval \([0,1]\), such that \( f(\pi) = 1 \) and \( f(2\pi) = 2 \). Determine the solution set of the system of inequalities \(\left\{\begin{array}{l}1 \leqslant x \leqslant 2, \\ 1 \leqslant f(x) \leqslant 2\end{array}\right.\).

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