
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 $C: \frac {x^{2}}{a^{2}}+ \frac {y^{2}}{b^{2}}=1$ with $a > b > 0$ and its left and right foci denoted as $F_1$ and $F_2$ respectively, and whose eccentricity is $\frac {1}{3}$. Point $P$ lies on ellipse $C$, and the maximum area of triangle $PF_1F_2$ is $2\sqrt{2}$.  
$(1)$ Find the equation of ellipse $C$.  
$(2)$ Given that the line $l: y = kx + 2$ ($k \neq 0$) intersects the ellipse $C$ at two distinct points $M$ and $N$. If there exists a point $G$ on the x-axis such that $|GM| = |GN|$, find the range for the x-coordinate of point $G$.

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