
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 $a > 0$ and $a \neq 1$. Define the function $f(x) = \begin{cases} a^{|x|}, & x < 1 \\ |x^2 - 2x|, & x \geqslant 1 \end{cases}$. Find the range of $a$ such that the solution set of the inequality $f(x) \leqslant 3$ is $(-\infty, 3]$.

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