
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(2, $\frac{\pi}{2}$), and B(2$\sqrt{2}$, $\frac{\pi}{4}$).  
(1) Find the polar equation of circle C<sub>1</sub> passing through points O, A, and B;  
(2) Establish a Cartesian coordinate system with the pole as the origin and the positive x-axis as the polar axis. The parametric equation of circle C<sub>2</sub> is $\begin{cases}x=-1+a\cos\theta \\ y=-1+a\sin\theta \end{cases}$ (where $\theta$ is the parameter). If circle C<sub>1</sub> is tangent to circle C<sub>2</sub> externally, find the value of the real number $a$.

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