
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 $\dfrac{x^2}{a^2} + \dfrac{y^2}{b^2} = 1 \left( a > b > 0 \right)$ with its left and right foci as $F_1$ and $F_2$, respectively, and its top vertex as $M$. If the slope of line $MF_1$ is $1$, and it intersects the ellipse at another point $N$, and the perimeter of $\triangle F_2MN$ is $4\sqrt{2}$.

$(1)$ Find the standard equation of the ellipse;

$(2)$ A line passing through point $F_1$ (the slope of line $l$ is not $1$) intersects the ellipse at points $P$ and $Q$, with point $P$ above point $Q$. If ${S_{\Delta F_1NQ}} = \dfrac{2}{3}{S_{\Delta F_1MP}}$, find the slope of line $l$.

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