
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
Find the value of \( f(2019) \), given that \( f(x) \) satisfies the following three conditions:
1. \( f(x) > 0 \) for any \( x > 0 \)
2. \( f(1) = 1 \)
3. \( f(a+b) \cdot (f(a) + f(b)) = 2 f(a) \cdot f(b) + a^2 + b^2 \) for any \( a, b \in \mathbb{R} \)

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