
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 line $ax - by + 2 = 0$ ($a > 0$, $b > 0$) intersects the circle C: $x^2 + y^2 + 2x - 2y = 0$ at two points A and B. When $|AB|$ is at its maximum, the minimum value of $\frac{1}{a} + \frac{4}{b}$ is $\_\_\_\_\_\_$.

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