
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 circles $M: 2x^{2}+2y^{2}-8x-8y-1=0$ and $N: x^{2}+y^{2}+2x+2y-6=0$, and 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 point $A$ is taken on line $l$ such that $\angle BAC = 45^{\circ}$, where $AB$ passes through the center of circle $M$ and $B, C$ are points on circle $M$.
$\qquad$(a) Find the equation of line $AC$ when the x-coordinate of point $A$ is 4.
$\qquad$(b) Find the range of possible x-coordinates for point $A$.

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