
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
The function sequence \(\left\{f_{n}(x)\right\}\) is recursively defined by the following conditions:
\[
\left\{\begin{array}{l}
f_{1}(x)=\sqrt{x^{2}+48} \\
f_{n+1}(x)=\sqrt{x^{2}+6 f_{n}(x)}
\end{array} \text{ for each } n \geq 1 .\right.
\]

Find all real solutions to the equation \(f_{n}(x)=2 x\).

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