
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 $C$: $\dfrac{x^{2}}{a^{2}} + \dfrac{y^{2}}{b^{2}} = 1 (a > b > 0)$, whose left and right foci are $F_{1}(-1,0)$ and $F(1,0)$ respectively. A line $l$ passing through $F_{1}$ intersects the ellipse $C$ at points $A$ and $B$, and the perimeter of $\triangle ABF_{2}$ is $4\sqrt{2}$.

(Ⅰ) Find the equation of the ellipse $C$;

(Ⅱ) Draw a line $m$ parallel to line $l$ passing through the point $(4,0)$, and let line $m$ intersect the parabola $y^{2} = 4x$ at points $P$ and $Q$. If $A$ and $P$ are above the $x$-axis, and line $PA$ intersects line $QB$ at a point $M$ on the $x$-axis, find the equation of line $l$.

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