
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 $\frac{x^{2}}{a^{2}}+ \frac{y^{2}}{b^{2}}=1 (a>b>0)$ on which the maximum distance from a point to the right focus $F(c, 0)$ is $\sqrt{2}+1$. Also, $1, \sqrt{2}a, 4c$ form a geometric sequence.
(I) Find the equation of the ellipse.
(II) A line $l$ passing through point $F$ and not perpendicular to the $x$-axis intersects the ellipse at points $A$ and $B$. The perpendicular bisector of line segment $AB$ intersects the $x$-axis at point $M(m, 0)$. Find the range of real number values for $m$.

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