
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 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 $\rho=4\cos \theta$.

$(I)$ Identify the type of curve $C_{1}$ and convert its equation 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 curves $C_{1}$ and $C_{2}$ lie on $C_{3}$, find $a$.

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