
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 coordinate system $xOy$, the equation of curve $M$ is $y=\sqrt{-{x}^{2}+4x}$, and the equation of curve $N$ is $xy=9$. Establish a polar coordinate system with the origin $O$ as the pole and the positive half-axis of the $x$-axis as the polar axis.<br/>$(1)$ Find the polar coordinate equations of curves $M$ and $N$;<br/>$(2)$ If the ray $l:\theta =\theta _{0}(\rho \geqslant 0$,$0 \lt \theta _{0} \lt \frac{π}{2})$ intersects curve $M$ at point $A$ (distinct from the pole) and intersects curve $N$ at point $B$, and $|OA|\cdot |OB|=12$, find $\theta _{0}$.

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