
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 right focus $F_2$ of the ellipse $C: \frac{x^{2}}{a^{2}}+ \frac{y^{2}}{b^{2}}=1 (a > b > 0)$ intersects the ellipse at points $A$ and $B$. Let $F_1$ be the left focus, the perimeter of $\triangle AF_{1}B$ is $4\sqrt{3}$, and the eccentricity of the ellipse is $\frac{\sqrt{6}}{3}$.
(1) Find the equation of ellipse $C$;
(2) Let $P$ be the lower vertex of the ellipse $C$, and the ellipse $C$ intersects with the line $y=kx+m$ at two distinct points $M$ and $N$. When $|PM|=|PN|$, find the range of real number values for $m$.

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