
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 function $f(x)=-x^{3}+3x+2$, suppose it takes its minimum value at $x\_1$ and maximum value at $x\_2$. The coordinates of points $A$ and $B$ on the $xy$-plane are $(x\_1,f(x\_1))$ and $(x\_2,f(x\_2))$, respectively. On the same plane, a moving point $P$ satisfies $\overrightarrow{PA}\cdot \overrightarrow{PB}=4$. Let point $Q$ be the reflection of point $P$ across the line $y=2(x-4)$. Find:

1. The coordinates of points $A$ and $B$;
2. The trajectory equation of point $Q$.

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