
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 a movable point $P(x, y)$ that satisfies the following conditions:
\[
\left\{
\begin{array}{l}
2x + y \leq 2, \\
x \geq 0, \\
\left( x + \sqrt{x^2 + 1} \right) \left( y + \sqrt{y^2 + 1} \right) \geq 1,
\end{array}
\right.
\]
find the area of the region formed by the point $P(x, y)$.

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