
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 $0 \leq y_3 \leq y_2 \leq y_1 \leq 1$ and $(1-y_1)^2 + 2(y_1-y_2)^2 + 2(y_2-y_3)^2 + y_3^2 = \frac{1}{2}$, find $y_1$.

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