
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
Let be the set  $ \mathcal{C} =\left\{ f:[0,1]\longrightarrow\mathbb{R}\left| \exists f''\bigg|_{[0,1]} \right.\quad\exists x_1,x_2\in [0,1]\quad x_1\neq x_2\wedge \left( f\left( 
x_1 \right) = f\left( x_2 \right) =0\vee  f\left( 
x_1 \right) = f'\left( x_1 \right) = 0\right)  \wedge f''<1 \right\} , $ and  $ f^*\in\mathcal{C} $  such that  $ \int_0^1\left| f^*(x) \right| dx =\sup_{f\in\mathcal{C}} \int_0^1\left| f(x) \right| dx .  $ Find  $ \int_0^1\left| f^*(x) \right| dx $  and describe  $ f^*. $ 

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