
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 that $F_{1}(-1,0)$ and $F_{2}(1,0)$ are respectively the left and right foci of the ellipse $G$: $\dfrac{x^{2}}{a^{2}}+ \dfrac{y^{2}}{b^{2}}=1(a > b > 0)$. Point $P$ is on the ellipse, and $PF_{2}\perp F_{1}F_{2}$, $|PF_{1}|-|PF_{2}|= \dfrac{a}{2}$.  
$(1)$ Find the equation of the ellipse $G$;  
$(2)$ If point $B$ is the top vertex of the ellipse $G$, and a line $l$ passing through $F_{2}$ intersects the ellipse $G$ at two distinct points $M$, $N$, is there a line $l$ such that the ratio of the areas of $\triangle BF_{2}M$ and $\triangle BF_{2}N$ is $2$? If it exists, find the equation of line $l$; if not, explain why.

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