
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
Ellipse $C_1$: $\frac{x^2}{2} + y^2 = 1$, Ellipse $C_2$: $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$ ($a > b > 0$) has one of its foci at $(\sqrt{5}, 0)$. A line $l$ with a slope of $1$ intersects ellipse $C_2$ at points $A$ and $B$, and the midpoint $H$ of segment $AB$ is at $(2, -1)$.

(1) Find the equation of ellipse $C_2$.

(2) Let $P$ be a point on ellipse $C_2$, and points $M$, $N$ on ellipse $C_1$, with $\overrightarrow{OP} = \overrightarrow{OM} + 2\overrightarrow{ON}$. Is the product of the slopes of lines $OM$ and $ON$ a constant? If yes, find the constant; if no, explain why.

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