
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 rectangular coordinate system $xOy$, the parametric equations of the curve $C$ are given by $\begin{cases} x=2+\cos\alpha \\ y=3+\sin\alpha \end{cases}$, where $\alpha$ is the parameter. Establish a polar coordinate system with the coordinate origin $O$ as the pole and the positive semi-axis of $x$ as the polar axis. Given that the polar coordinates of point $A$ are $(3, \frac{\pi}{2})$.

1. Find the polar equation of the curve $C$.
2. Draw a tangent line to the curve $C$ passing through point $A$, with $M$ as the point of tangency. Draw another tangent line to the curve $C$ passing through the origin $O$, with $N$ as the point of tangency. Calculate the ratio $\frac{|ON|}{|AM|}$.

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