
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
On the circle $\left(x-4\right)^{2} + \left(y-3\right)^{2} = r^{2} (r > 0)$, there always exists a point $P$ such that two mutually perpendicular tangents can be drawn through point $P$ to the ellipse ${x^2} + \frac{y^2}{3} = 1$. Determine the possible range of values for $r$.

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