
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 $f(x)=\ln x$, the curve $y=g(x)$ is symmetric to the curve $y=f(x)$ about the line $y=x$. If there exists a line passing through the origin that is tangent to both the curve $y=f(x)$ and the curve $y=g(ax)$, then the value of the real number $a$ is \_\_\_\_\_\_.

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