
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 C1: $x^2 + y^2 + 2ax + a^2 - 4 = 0$ (where $a \in \mathbb{R}$) and C2: $x^2 + y^2 - 2by - 1 + b^2 = 0$ (where $b \in \mathbb{R}$) are externally tangent to each other, find the maximum value of $a + b$.

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