
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 $(C)$: $\frac{x^{2}}{a^{2}}+ \frac{y^{2}}{b^{2}}=1(a > b > 0)$, its left and right foci are $F_{1}$, $F_{2}$ respectively. A line passing through $F_{2}$ and perpendicular to the $x$-axis intersects the ellipse $(C)$ at points $A$ and $B$ ($|AB|= \frac{8 \sqrt{5}}{5}$). Point $P$ is a moving point on the ellipse $(C)$, and the minimum value of $\cos ∠F_{1}PF_{2}$ is $\frac{3}{5}$.

1. Find the equation of the ellipse $(C)$;
2. A line $(l)$ passing through the point $(-2,0)$ intersects the ellipse at points $M$ and $N$. Find the range of values for $\overrightarrow{F_{2}M} \cdot \overrightarrow{F_{2}N}$.

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