
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 lines $l\_1$: $y=k\_1x+3$ and $l\_2$: $y=k\_2x-3$, where $k\_1 \cdot k\_2=-\frac{9}{16}$. The lines intersect at point $P$, and there is another point $Q(a,0)$.

1. Find the equation of the trajectory $C$ of point $P$.
2. If the length of $PQ$ is always minimized when point $P$ is on the $x$-axis, find the range of $a$.
3. If $a \in (0, 4)$, draw a vertical line from $Q(a,0)$ to the $x$-axis, intersecting trajectory $C$ at points $M$ and $N$. Draw lines $AM$ and $BN$ with $A(-4,0)$ and $B(4,0)$. Draw a line perpendicular to $AM$ through $Q$ that intersects line $BN$ at point $E$. Find the ratio of the areas of $\triangle BQE$ and $\triangle BQN$.

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