
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 $\mathbb{R}$ with a period of 2, which is monotonically decreasing on the interval $[0, 1]$ and satisfies $f(\pi) = 1$, $f(2\pi) = 2$. Determine the solution set of the inequality system:

$$\begin{cases} \overset{1\leq x\leq 2}{1\leq f(x)\leq 2}\end{cases}$$

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