
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 that the right focus of the ellipse $\frac{{x}^{2}}{{a}^{2}}+\frac{{{y}^{2}}}{{{b}^{2}}}=1(a > b > 0)$ is $(2\sqrt{2},0)$, and the sum of the distances from a point $M$ on the ellipse to its two foci ${F}_{1}$, ${F}_{2}$ is $4\sqrt{3}$.

(1) Find the equation of the ellipse;

(2) Let line $l$: $y=x+m(m∈R)$ intersect the ellipse at two distinct points $A$, $B$, with $|AB|=3\sqrt{2}.$ If point $P(x_{0},2)$ satisfies $||\overrightarrow{PA}||=||\overrightarrow{PB}||$, find the value of $x_{0}$.

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