
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 an ellipse $C$: $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1\left(a \gt b \gt 0\right)$ with left and right foci $F_{1}$ and $F_{2}$, respectively, and $|F_{1}F_{2}|=2\sqrt{2}$. Let $M$ be a point on the ellipse, and the line segment $MF_{1}$ is tangent to the circle $C$: $x^{2}+y^{2}=1$ at its midpoint $N$.  
$(1)$ Find the equation of the ellipse $C$.  
$(2)$ If a line $l$ passing through $F_{1}$ intersects the ellipse $C$ at points $A$ and $B$, and there exists a point $P$ on the ellipse $C$ such that $\overrightarrow{OP}=\overrightarrow{OA}+\overrightarrow{OB}$, find the equation of the line $l$.

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