
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 $C$ with its center at the origin $O$, its foci on the $x$-axis, and it passes through points $A_{1}(-2,0)$ and $A_{2}(\sqrt{2}, \frac{\sqrt{2}}{2})$

(Ⅰ) Find the standard equation of the ellipse $C$;  
(Ⅱ) A line $l$ passing through the focus $F$ of the parabola $y^{2}=4x$ intersects the ellipse $C$ at two distinct points $M$ and $N$, and satisfies $\overrightarrow{OM} \perp \overrightarrow{ON}$. Find the equation of line $l$.

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