
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 parametric equation of circle $C$ is given by $\begin{cases} x = 5\cos \alpha \\ y = -6 + 5\sin \alpha \end{cases}$ (with $\alpha$ as the parameter). Taking the origin as the pole and the positive x-axis as the polar axis, establish the polar coordinate system.
$(1)$ Find the polar equation of circle $C$.
$(2)$ The polar equation of line $l$ is $\theta = \alpha_0$, where $\alpha_0$ satisfies $\tan \alpha_0= \frac { \sqrt {5}}{2}$. $l$ intersects $C$ at points $A$ and $B$. Calculate the length of $|AB|$.

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