
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
Assume \( f_{1}(x)=\sqrt{x^{2}+32} \) and \( f_{n+1}(x)=\sqrt{x^{2}+\frac{16}{3} f_{n}(x)} \) for \( n=1,2, \cdots \). For each \( n \), find the real solution of \( f_{n}(x)=3 x \).

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