
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
Determine the equations of the circles under the following conditions:
1. The circle passes through the points $(0,4)$ and $(4,6)$, and its center lies on the line $x-2y-2=0$;
2. The radius of the circle is $\sqrt{13}$, and it is tangent to the line $2x+3y-10=0$ at the point $(2,2)$.

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