
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
When trekking through the Enchanted Forest, Alice finds a magical gate whose boundary is described by the equation $x^2 + y^2 + 2x - 4y = -3$. She needs to know the diameter of the gate to determine if her group can pass through it together. What is the diameter of the gate?

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