
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 a circle $M$ with equation $(x-1)^2 + (y-1)^2 = 4$, and a line $l$ with equation $x + y - 6 = 0$. Let $A$ be a point on line $l$, and suppose there exist two points $B$ and $C$ on circle $M$ such that $\angle BAC = 60^\circ$. Determine the range of the $x$-coordinate $x_0$ of point $A$.

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