
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 an ellipse C with an eccentricity of $\frac{\sqrt{2}}{2}$. Let A and B be its left and right vertices, respectively, and $F_2$ be its right focus. P is a point on the ellipse C, different from A and B, and the minimum value of the dot product of $\vec{PA}$ and $\vec{PB}$ is -2.  
(1) Find the equation of the ellipse C;  
(2) If a line passing through the left focus $F_1$ intersects the ellipse at points M and N, find the range of values for the dot product of $\vec{F_{2}M}$ and $\vec{F_{2}N}$.

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