
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 $C$: $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 (a > b > 0)$ with a focal distance of $2\sqrt{3}$, a line with a slope of $\frac{1}{2}$ intersects the ellipse at points $A$ and $B$. If the midpoint of segment $AB$ is $D$, and the slope of line $OD$ is $-\frac{1}{2}$,

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

$(2)$ If a line $l$ with slope $k$ passes through the left focus $F$ and intersects the ellipse at points $M$ and $N$, and point $P$ is on the ellipse satisfying $OP \perp MN$, is $\frac{1}{|MN|} + \frac{1}{|OP|^2}$ a constant? If so, find this constant; if not, explain why.

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