
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]


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In the rectangular coordinate system $(xOy)$, the polar coordinate equation of curve $C_{1}$ is $\rho^{2} - 2\rho \sin \theta + 1 - a^{2} = 0 (a > 0)$. In the polar coordinate system with the coordinate origin as the pole and the positive semi-axis of $x$ as the polar axis, the curve $C_{2}$ is $\rho = 4\cos \theta$.

(1) Explain what kind of curve $C_{1}$ is and convert its equation into rectangular coordinates;
(2) The polar coordinate equation of line $C_{3}$ is $\theta = \alpha_{0}$, where $\tan \alpha_{0} = 2$. If the common points of curves $C_{1}$ and $C_{2}$ both lie on $C_{3}$, find $a$.

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