
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
Solve the system of equations:
$$
\left\{
\begin{array}{l}
\frac{1}{\sqrt{1 + 2x^2}} + \frac{1}{\sqrt{1 + 2y^2}} = \frac{2}{\sqrt{1 + 2xy}}, \\
\sqrt{x(1 - 2x)} + \sqrt{y(1 - 2y)} = \frac{2}{9}
\end{array}
\right.
$$

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