
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
In the Cartesian coordinate system $xOy$, for $P(a,b)$ and point $Q(a,b')$, the following definition is given: if $b' = \left\{\begin{array}{l}b & (a \geqslant 1)\\-b & (a < 1)\end{array}\right.$, then point $Q$ is called the limit point of point $P$. For example, the coordinates of the limit point of point $(2,3)$ are $(2,3)$, and the coordinates of the limit point of point $(-2,5)$ are $(-2,-5)$.

(1) The coordinates of the limit point of $(\sqrt{3},1)$ are ____;

(2) Determine which of the points $A(-2,-1)$ and $B(-1,2)$ is the limit point of a point on the graph of the function $y = \frac{2}{x}$, and explain the reason;

(3) If the point $P(a,b)$ is on the graph of the function $y = -x+3$, and the range of the ordinate of its limit point $Q(a,b')$ is $-6 \leqslant b' \leqslant -3$, find the range of $a$.

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