
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, and a strictly increasing function on the interval $[-1, 0]$. Given that $f(e) = 0$ and $f(2e) = 1$, find the solution set of the inequality:

$$\begin{cases} 0 \leq x \leq 1 \\ 0 \leq f(x) \leq 1 \end{cases}$$

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