
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
In the Cartesian plane with coordinates $(x, y)$, a circle $M$ is defined by the equation $((x-2)^2+y^2=1)$, and a curve $C$ is given by the parametric equations $x=\cos \alpha, y=\sin \alpha$ with $\alpha$ as the parameter. In the polar coordinate system with the origin as the pole and the positive $x$-axis as the polar axis, the polar equation of a line $l$ is given by $\theta=\frac{\pi}{6}$. Suppose that $l$ is tangent to the circle $M$ at point $A$ and intersects curve $C$ at point $N$ in the third quadrant, find the area of triangle $△AN$.

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