
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 ellipse $E$: $\frac{x^{2}}{a^{2}} + \frac{y^{2}}{b^{2}} = 1 (a > b > 0)$ passing through the point $(4,0)$ and sharing the same foci with the hyperbola $x^{2} - y^{2} = 2$.

(1) Find the standard equation of ellipse $E$;
(2) Let point $M(m,0)$ be on the major axis of ellipse $E$, and point $P$ be any point on the ellipse. When $| \overrightarrow{MP} |$ is minimized, point $P$ coincides with the right vertex of the ellipse. Find the range of values for the real number $m$.

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