
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 the parabola $y=\frac{1}{4}x^2$ and the circle $C: (x-1)^2+(y-2)^2=r^2 (r > 0)$ with a common point $P$. If the tangent line to the parabola at point $P$ also tangents the circle $C$, find the value of $r$.

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