
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 $G$: $\frac{x^{2}}{a^{2}}+ \frac{y^{2}}{b^{2}}=1(a > b > 0)$ with an eccentricity $e= \frac{\sqrt{3}}{3}$, and its right vertex at $(\sqrt{3},0)$.
(1) Find the equation of $G$;
(2) The line $y=kx+1$ intersects the curve $G$ at two distinct points $A$ and $B$. If there exists a point $M$ on the $x$-axis such that $|AM|=|BM|$, find the range of values for the $x$-coordinate of point $M$.

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