
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
Consider the functions \( f(x) \) that are defined in the interval \((0,1)\), are positive, integrable, have an integral value of 1 on the segment \((0,1)\), and are concave from below. (This latter condition means that for any \( 0 \leq x_{1} < x_{2} \leq 1 \) and \( 0 \leq \lambda \leq 1 \),

\[ 
f\left(\lambda x_{1}+(1-\lambda) x_{2}\right) \geq \lambda f\left(x_{1}\right)+(1-\lambda) f\left(x_{2}\right)
\]

is satisfied.)

What values can the following integral take among these functions?

\[ 
\int_{0}^{1} f^{2}(x) \, dx 
\]

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