
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 that $P(x_{0},y_{0})$ is a point outside the circle $C$: $x^{2}+(y-4)^{2}=1$, a tangent line is drawn from point $P$ to circle $C$, with the tangent points being $A$ and $B$. Let $f(P)$ denote the area of the quadrilateral $PACB$. When $P(x_{0},y_{0})$ moves along circle $D$: ${{(x+4)}^{2}}+{{(y-1)}^{2}}=4$, determine the range of values for $f(P)$.

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