
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 the function $f(x) = ax^2 + c$ ($a \neq 0$). If $\int_{0}^{1} f(x) dx = f(x_0)$, where $0 \leq x_0 \leq 1$, then the value of $x_0$ is ______.

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