
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 the equation of curve $C$ is: $ax^{2}+ay^{2}-2a^{2}x-4y=0$ ($a\neq 0$, $a$ is a constant).

(Ⅰ) Determine the shape of curve $C$;  
(Ⅱ) Suppose line $l$: $y=-2x+4$ intersects curve $C$ at two distinct points $M$ and $N$, and $|OM|=|ON|$, find the equation of curve $C$.

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