
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 a function \( f(x) \) that satisfies:
\[ f(m+n) = f(m) f(n), \quad f(1) = 3 \]
find the value of 
\[ \sum_{k=1}^{4} \frac{f^{2}(k)+f(2k)}{f(2k-1)} \]

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