
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 $E$: $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1(a>b>0)$ with the upper vertex $B$ and the left focus $F$, and $B$ and $F$ lie on the line $x-y+2=0$. <br/>$(1)$ Find the standard equation of $E$;<br/>$(2)$ Let the line $l$ intersect $E$ at points $P$ and $Q$, and the quadrilateral $BPFQ$ is a parallelogram. Find the equation of $l$.

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