
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 $C_1$ is described by the parametric equations $\begin{cases} x=a\cos t \\ y=1+a\sin t \end{cases}$ ($t$ 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 curve $C_2$ is described by $\rho =4\cos \theta$.

$(1)$ Identify the type of curve $C_1$ is and convert its equation into a polar coordinate equation;

$(2)$ The polar coordinate equation of line $C_3$ is $\theta ={\sigma_0}$, where ${\sigma_0}$ satisfies $\tan {\sigma_0}=2$. 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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