
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 $C: x^{2}+y^{2}-2x+4y-4=0$, circle $C_{1}$: $\left(x-3\right)^{2}+\left(y-1\right)^{2}=4$, and point $P(3,1)$.<br/>$(Ⅰ)$ Determine the position relationship between circle $C$ and circle $C_{1}$;<br/>$(Ⅱ)$ Find the equation of the line passing through point $P$ and tangent to circle $C$.

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