
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\in\mathcal{C}^1 [0,1] $  that satisfy  $ f(1)=-1/6 $  and  $$  \int_0^1 \left( f'(x) \right)^2 dx\le 2\int_0^1 f(x)dx.  $$  

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