
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 $C$: $\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1$ with an eccentricity of $\frac{\sqrt{2}}{2}$, a focal length of $2$, and a right focus point $F$. The line passing through point $F$ intersects the ellipse at points $A$ and $B$.

1. Find the equation of ellipse $C$.
2. Determine if there exists a fixed point $M$ on the $x$-axis such that $\overrightarrow{MA}\cdot\overrightarrow{MB}$ is constant. If it exists, find the constant value and the coordinates of point $M$; otherwise, explain the reason.

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