
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 ellipse $C$: $\frac{{{x}^{2}}}{{{a}^{2}}}+\frac{{{y}^{2}}}{{{b}^{2}}}=1(a > b > 0)$ with left and right foci $F\_{1}$, $F\_{2}$, and points $A(a,0)$, $B(0,b)$ satisfying the condition $|AB|=\frac{\sqrt{2}}{2}|{{F}\_{1}}{{F}\_{2}}|$.

(1) Find the eccentricity of ellipse $C$;

(2) If the distance from the coordinate origin $O$ to line $AB$ is $\frac{3\sqrt{3}}{2}$, find the equation of ellipse $C$;

(3) Under the conditions of (2), line $l$ passes through point $P(-2,1)$ and intersects ellipse $C$ at points $M$ and $N$. Point $P$ is the midpoint of segment $MN$. Find the equation of line $l$.

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