
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
Given circle $O_1: x^{2}+(y+1)^{2}=4$ and the center of circle $O_2$ is at $(3,3)$, and the two circles are externally tangent to each other, find:  
$(1)$ The standard equation of circle $O_2$;  
$(2)$ The general equation of the common internal tangent lines of the two circles.

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