
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 differentiable functions  $ f:\mathbb{R}\longrightarrow\mathbb{R} $  that verify the conditions: $ \text{(i)}\quad\forall x\in\mathbb{Z} \quad f'(x) =0 $  $ \text{(ii)}\quad\forall x\in\mathbb{R}\quad f'(x)=0\implies f(x)=0 $  

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