
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 \( f(x) \) such that \( f(0) = 1 \), \( f\left(\frac{\pi}{2}\right) = 2 \), and for any \( x, y \in \mathbf{R} \), the following inequality holds:
\[ 
f(x+y) + f(x-y) \leq 2 f(x) \cos y 
\]

Find \( f(x) \).

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