
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
In the Cartesian coordinate plane $(xOy)$, find the standard equation of the circle with the largest radius among all circles that are tangent to the line $2mx - y - 4m + 1 = 0 (m \in \mathbb{R})$ and have their center at point $(1, 0)$.

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