
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 polar coordinate system, three points are given: $O(0, 0)$, $A\left(2, \frac{\pi}{2}\right)$, and $B\left(2\sqrt{2}, \frac{\pi}{4}\right)$.

$(1)$ Find the polar equation of the circle $C_1$ passing through points $O$, $A$, and $B$;

$(2)$ Set up a Cartesian coordinate system with the pole as the origin and the polar axis as the positive half of the $x$-axis. The parametric equation of circle $C_2$ is $\begin{cases} x = -1 + a \cos \theta, \\ y = -1 + a \sin \theta \end{cases}$ (where $\theta$ is the parameter). If circle $C_1$ is externally tangent to circle $C_2$, find the value of the real number $a$.

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