
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 an eccentricity of $\frac{\sqrt{2}}{2}$ and passes through the point $P(2,2)$.
(1) Find the standard equation of the ellipse $C$;
(2) A line $l$ passing through $M(-1,0)$ intersects the ellipse $C$ at points $A$ and $B$. The left and right foci of the ellipse $C$ are $F_{1}$ and $F_{2}$, respectively. Let $S_{1}$ and $S_{2}$ denote the areas of triangles $\triangle F_{1}AF_{2}$ and $\triangle F_{1}BF_{2}$, respectively. Determine the range of values for $|S_{1}-S_{2}|$.

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