
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
Given the function $f(x)$ satisfy $f(-x) = f(x)$, and for $x_{1}$, $x_{2} \in [0, +\infty)$, $\frac{{f(x_{1}) - f(x_{2})}}{{x_{1} - x_{2}}} > 0$. For any $x \in [\frac{1}{2}, 1]$, the inequality $f(ax+1) \leq f(x-2)$ always holds, determine the range of real number $a$.

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