
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 \( f_{1}(x)=\sqrt{x^{2}+48} \), and for \( n \in \mathbb{Z}_{+}, n \geq 2 \):
\[ f_{n}(x)=\sqrt{x^{2}+6 f_{n-1}(x)} \]

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

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