
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
On the Cartesian plane consider the set  $V$  of all vectors with integer coordinates. Determine all functions  $f : V \rightarrow \mathbb{R}$  satisfying the conditions:
(i)  $f(v) = 1$  for each of the four vectors  $v \in V$  of unit length.
(ii)  $f(v+w) = f(v)+f(w)$  for every two perpendicular vectors  $v, w \in V$ 
(Zero vector is considered to be perpendicular to every vector).

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