
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 $\frac {x^{2}}{a^{2}} + \frac {y^{2}}{b^{2}} = 1$ ($a > b > 0$) with eccentricity $\frac { \sqrt {2}}{2}$, the maximum distance from any point on the ellipse to the right focus $f$ is $\sqrt {2}+1$.  
(I) Find the equation of the ellipse;  
(II) Given that point $C(m, 0)$ is a fixed point on the line segment $OF$ different from $O$ and $F$ ($O$ is the origin), determine whether there exists a line $l$ passing through point $F$ and not perpendicular to the x-axis, intersecting the ellipse at points $A$ and $B$, such that $|AC| = |BC|$, and explain the reason.

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