
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
For $a, b \in \mathbf{R}$, we define
$$
\max \{a, b\}=\left\{\begin{array}{ll}
a, & a \geqslant b; \\
b, & a < b,
\end{array}\right.
$$

The function $f(x) = \max \left\{2^{-x}, -|x-1| + 2\right\}$. Determine the range of real values $a$ for which the equation $f(x) = a$ has three roots.

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