
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
The circles $x^2+y^2+2ax+a^2-9=0$ and $x^2+y^2-4by-1+4b^2=0$ have three common tangents. Given $a\in \mathbb{R}$, $b\in \mathbb{R}$, and $ab\neq 0$, find the minimum value of $\frac {4}{a^{2}}+ \frac {1}{b^{2}}$.

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