
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 $C_1: x^2+y^2+2ax+a^2-4=0$ and $C_2: x^2+y^2-2by+b^2-1=0$ are internally tangent, where $a, b \in \mathbb{R}$ and $ab \neq 0$, find the minimum value of $\frac{1}{a^2} + \frac{1}{b^2}$.

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