
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
Elective 4-4: Coordinate System and Parametric Equations

In the Cartesian coordinate system $xOy$, the curve $C$ is given by the parametric equations $\begin{cases} & x=-6+5\cos \theta, \\ & y=5\sin \theta \\ \end{cases}$ ($\theta$ is the parameter).

$(1)$ Establish a polar coordinate system with the origin as the pole and the non-negative half-axis of $x$ as the polar axis, and find the polar equation of curve $C$;

$(2)$ The line $l$ is given by $\begin{cases} & x=t\cos \alpha, \\ & y=t\sin \alpha \\ \end{cases}$ ($t$ is the parameter), and line $l$ intersects curve $C$ at points $A$ and $B$, with $|AB|=2\sqrt{7}$. Find the inclination angle $\alpha$ of line $l$.

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