
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 $C\_1$: $x^{2}+y^{2}-6x+5=0$, parabola $C\_2$: $y^2=x$, and line $l$ passing through point $M(m,0)$. Line $l$ intersects circle $C\_1$ at points $A$ and $B$, and intersects parabola $C\_2$ at points $C$ and $D$.
(1) If $m=0$, find the trajectory equation of the midpoint $R$ of segment $AB$ as line $l$ rotates around point $M$;
(2) If $m=2$ and $\overrightarrow{AC} = \overrightarrow{DB}$, find the equation of line $l$.

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