
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 $f(x)=\log_{3}\frac{x}{3} \cdot \log_{3}\frac{x}{27}$, if $f(x_{1})=f(x_{2})$ holds for distinct positive numbers $x_{1}$ and $x_{2}$, then the minimum value of $\frac{1}{x_{1}} + \frac{9}{x_{2}}$ is ______.

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