
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 that two circles $x^2+y^2+2ax+a^2-4=0$ and $x^2+y^2-4by-1+4b^2=0$ have exactly three common tangents, where $a, b \in \mathbb{R}, ab \neq 0$, the minimum value of $\frac{4}{a^2} + \frac{1}{b^2}$ is \_\_\_\_\_\_.

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