
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
For any point on the circle $x^2 + y^2 + 2x - 4y + 1 = 0$, its symmetric point with respect to the line $2ax - by + 2 = 0$ ($a, b \in \mathbb{R}^+$) still lies on the same circle. Find the minimum value of $\frac{1}{a} + \frac{2}{b}$.

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