
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 a function $f(x)$ satisfying $f(x+3) = 3f(x)$ for all $x$ and $f(x) = \ln(x) - ax$ when $x \in (0, 3)$, where $a > \frac{1}{3}$. Find the value of the real number $a$ if the maximum value of $f(x)$ on the interval $(-6, -3)$ is $-\frac{1}{9}$.

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