
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 $xy$, the parametric equation of curve $C_1$ is
\[
\begin{cases}
x=a\cos t \\
y=1+a\sin t
\end{cases}
\]
where $t$ is the parameter, $a > 0$. In the polar coordinate system with the origin as the pole and the positive $x$-axis as the polar axis, the equation of curve $C_2$ is $\rho=4\cos \theta$.

$(I)$ Explain what kind of curve $C_1$ is and convert the equation of $C_1$ into a polar coordinate equation;

$(II)$ The polar coordinate equation of line $C_3$ is $\theta=\alpha_{0}$, where $\alpha_{0}$ satisfies $\tan \alpha_0=2$. If all the common points of curve $C_1$ and $C_2$ lie on $C_3$, find $a$.

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