
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 two circles: $x^{2}+y^{2}+2ax+a^{2}-9=0$ and $x^{2}+y^{2}-4by-1+4b^{2}=0$, they have three shared tangents. If $a\in R$, $b\in R$, and $ab\neq 0$, determine 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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