
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
Given that $0 \leq x_3 \leq x_2 \leq x_1 \leq 1$ and $(1-x_1)^2 + 2(x_1-x_2)^2 + 2(x_2-x_3)^2 + x_3^2 = \frac{1}{2}$, find $x_1$.

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