
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
Consider the line $l: x + y - 4 = 0$ and the circle $x^2 + y^2 = 4$. Let point $A$ lie on line $l$. If there exist two points $B$ and $C$ on the circle such that $\angle BAC = 60^\circ$, find the maximum possible value of the $x$-coordinate of point $A$.

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