
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 a line $l$ intersects with the parabola $y^2=-x$ at points $A$ and $B$. The projections of $A$ and $B$ on the directrix are $A_1$ and $B_1$, respectively.  
(Ⅰ) If the midpoint of segment $AB$ has coordinates $(-4, 1)$, find the equation of line $l$;  
(Ⅱ) If the equation of line $l$ is $x=my-1$, where $m\in\mathbb{R}$, find the area of the trapezoid $AA_1B_1B$ (expressed in terms of $m$).

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