
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 circle $C: x^{2}+\left(y-1\right)^{2}=6$, where $AB$ are two moving points on circle $C$ with $AB=2\sqrt{2}$, and $G$ is the midpoint of chord $AB$. On the line $l: x-y-2=0$, there are two moving points $PQ$ with $PQ=2$. As $AB$ moves on circle $C$, the angle $\angle PGQ$ is always acute. Find the range of the abscissa of the midpoint $M$ of segment $PQ$.

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