
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
Consider the ellipse $C\_1$: $\frac{x^{2}}{2}+y^{2}=1$ and the ellipse $C\_2$: $\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1$ ($a > b > 0$). One focus of ellipse $C\_2$ has the coordinates $(\sqrt{5},0)$. The line $l$ with slope $1$ intersects ellipse $C\_2$ at points $A$ and $B$, and the midpoint $H$ of the line segment $AB$ has the coordinates $(2,-1)$.

1. Find the equation of ellipse $C\_2$.
2. Let $P$ be a point on the ellipse $C\_2$, and let $M$ and $N$ be points on the ellipse $C\_1$ such that $\overrightarrow{OP}=\overrightarrow{OM}+2\overrightarrow{ON}$. Determine whether the product of the slopes of lines $OM$ and $ON$ is constant. If it is, find the constant value; if not, explain the reason.

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