
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
Let point $F$ be the left focus of the hyperbola $C: \frac{{x}^{2}}{m+1}-\frac{{y}^{2}}{3-m}=1$. A line passing through the origin $O$ with a slope $k⩾\sqrt{3}$ intersects the hyperbola $C$ at points $A$ and $B$, where $P$ is the midpoint of $AF$ and $Q$ is the midpoint of $BF$. If $OP\bot OQ$, find the range of eccentricity $e$ of the hyperbola $C$.

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