
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 $C_{1}$: $x^{2}+y^{2}-2x-2y+1=0$, circle $C_{2}$: $\left(x-4\right)^{2}+\left(y-5\right)^{2}=r^{2}(r \gt 0)$.<br/>$(1)$ If circle $C_{1}$ and circle $C_{2}$ are externally tangent, find the value of $r$;<br/>$(2)$ If circle $C_{1}$ and circle $C_{2}$ intersect at two points, find the range of values for $r$.

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