
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 \( f(x) \) satisfies:
$$
\begin{array}{l}
f(1)=\frac{1}{4}, \\
4f(x)f(y)=f(x+y)+f(x-y) \quad \text{for} \quad x, y \in \mathbf{R},
\end{array}
$$

Determine \( f(2019) \).

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