
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 $\Gamma$: $\frac{x^{2}}{a^{2}} + \frac{y^{2}}{b^{2}} = 1$ ($a > b > 0$) with its right focus at $(2\sqrt{2}, 0)$, and passing through a point where $c > 1$.

(Ⅰ) Find the standard equation of the ellipse $\Gamma$;  
(Ⅱ) Let line $l: y = x + m$ ($m \in \mathbb{R}$) intersect the ellipse $\Gamma$ at two distinct points $A$ and $B$, and let $|AB| = 3\sqrt{2}$. 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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