
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 $C:\frac{{{x^2}}}{{{a^2}}}+\frac{{{y^2}}}{{{b^2}}}=1(a＞b＞0)$ with its left and right foci $F_{1}$ and $F_{2}$, and eccentricity $\frac{{\sqrt{3}}}{3}$. Point $P$ lies on the ellipse $C$, and the maximum area of triangle $\triangle PF_{1}F_{2}$ is $\sqrt{2}$.<br/>$(1)$ Find the equation of the ellipse $C$;<br/>$(2)$ Given a line $l:y=kx+1\left(k \gt 0\right)$ intersects the ellipse $C$ at two distinct points $M$ and $N$, where point $G(m,0)$ exists on the axis such that $|GM|=|GN|$, find the range of real number $m$.

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