
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 $M$: $\frac{x^{2}}{a^{2}} + \frac{y^{2}}{b^{2}} = 1 (a > b > 0)$ with a focal length of $2\sqrt{3}$ and an eccentricity of $\frac{\sqrt{3}}{2}$.
1. Find the equation of the ellipse $M$.
2. If a circle $N$: $x^2 + y^2 = r^2$ has a tangent line $l$ with slope $k$ that intersects ellipse $M$ at points $P$ and $Q$, can $OP$ and $OQ$ be perpendicular? If they can, find the corresponding value of $r$; if not, explain why.

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