
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
Given circle $C_{1}$: $x^{2}+\left(y-a\right)^{2}=a^{2}(a \gt 0)$, the distance from the center of the circle to the line $x-y-2=0$ is $2\sqrt{2}$. Find the number of common tangents of circle $C_{1}$ and circle $C_{2}$: $x^{2}+y^{2}-2x-4y+4=0$.

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