
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 point $P(4,4)$ and circle $C$: $(x-m)^2 + y^2 = 5$ $(m < 3)$, and ellipse $E$: $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$ $(a > b > 0)$ share a common point $A(3,1)$. $F_1$ and $F_2$ are the left and right foci of the ellipse, respectively, and line $PF_1$ is tangent to circle $C$.

(1) Find the value of $m$ and the equation of ellipse $E$,

(2) Let $Q$ be a moving point on ellipse $E$, find the range of values for $\overrightarrow{AP} \cdot \overrightarrow{AQ}$.

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