
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
Let $p$ be the set of positive rational numbers. Define the function $f: p \rightarrow p$ such that it satisfies the following conditions:

1. \( f(x) + f\left(\frac{1}{x}\right) = 1 \)
2. \( f(2x) = 2 f(f(x)) \) for all \( x \in p \).

Find the function \( f(x) \) and provide a justification.

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