
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 with the equation $\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1 (a > b > 0)$ and an eccentricity of $\frac{\sqrt{2}}{2}$. The right focus of the ellipse is $F(1,0)$. A line $l$ passes through point $F$ and intersects the ellipse at points $A$ and $B$. $O$ represents the origin of the coordinate system.

(I) Find the standard equation of the ellipse;

(II) As line $l$ rotates around point $F$, determine if there exists a fixed point $M$ on the $x$-axis such that $\overrightarrow{MA} \cdot \overrightarrow{MB}$ remains constant. If such a point exists, find its coordinates. If not, explain the reason.

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