
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 $f(x)=\log_{3}\left(\frac{3}{x}\right) \cdot \log_{3}\left(\frac{x}{27}\right)$.
$(1)$ Find the range of the function $f(x)$;
$(2)$ If the inequality $f(x)+5 \leq a - \sqrt{a}$ holds for all $x \in (0, +\infty)$, find the range of positive real number $a$.

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