
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 two circles M: $2x^2+2y^2-8x-8y-1=0$ and N: $x^2+y^2+2x+2y-6=0$, and the line $l: x+y-9=0$.  
(1) Find the equation of the circle passing through the intersection points of circles M and N, and the origin O.  
(2) A line passing through a point on line $l$ is drawn such that $\angle BAC=45^\circ$, with side AB passing through the center of circle M, and points B and C on circle M.  
① When the x-coordinate of point A is 4, find the equation of line AC.  
② Determine the range of the x-coordinate of point A.

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