
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 curve $C$: $\begin{cases}x=2\cos \alpha \\ y= \sqrt{3}\sin \alpha\end{cases}$ ($\alpha$ is a parameter) and the fixed point $A(0, \sqrt{3})$, $F_1$ and $F_2$ are the left and right foci of this curve, respectively. Establish a polar coordinate system with the origin $O$ as the pole and the positive half-axis of $x$ as the polar axis.

$(1)$ Find the polar equation of the line $AF_2$;

$(2)$ A line passing through point $F_1$ and perpendicular to the line $AF_2$ intersects this conic curve at points $M$ and $N$. Find the value of $||MF_1|-|NF_1||$.

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