
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 $F\_1(-c,0)$ and $F\_2(c,0)$ are the left and right foci of the ellipse $G: \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 (0 < b < a < 3)$, and point $P(2, \sqrt{2})$ is a point on ellipse $G$ with $|PF\_1| - |PF\_2| = a$.

(1) Find the equation of the ellipse $G$;

(2) Suppose line $l$ intersects ellipse $G$ at points $A$ and $B$, with $\overrightarrow{OA} \perp \overrightarrow{OB}$, where $O$ is the coordinate origin. Determine whether the distance from $O$ to line $l$ is a constant value. If it is, find this value; if not, explain the reason.

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