
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
Given the polar equation of curve $C$ is $\rho=4\cos \theta$. Taking the pole as the origin of the Cartesian coordinate system and the polar axis as the positive half-axis of $x$, establish the Cartesian coordinate system. The parametric equation of line $l$ is $\begin{cases}x=1+t\cos \alpha \\ y=t\sin \alpha\end{cases}$ (where $t$ is the parameter).

$(1)$ Convert the polar equation of curve $C$ into a Cartesian coordinate equation;

$(2)$ If line $l$ intersects curve $C$ at points $A$ and $B$, and $|AB|= \sqrt{14}$, find the value of the inclination angle $\alpha$ of the line.

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