
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 that the function $f(x)$ satisfies $f(x)=2f(\frac{1}{x})$, and $f(x) \neq 0$. When $x \in [1,3]$, $f(x)=\ln x$. If the function $g(x)=f(x)-ax$ has three distinct zeros in the interval $[\frac{1}{3},3]$, then the range of the real number $a$ is _______.

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