
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 all functions \( f: \mathbb{R} \rightarrow \mathbb{R} \) such that
(i) \( f(2u) = f(u+v) f(v-u) + f(u-v) f(-u-v) \) for all \( u, v \in \mathbb{R} \), and
(ii) \( f(u) \geq 0 \) for all \( u \in \mathbb{R} \).

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