
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 $C_1: x^2+y^2-6x+5=0$, and parabola $C_2: y^2=x$. A line $l$ passing through point $M(m,0)$ intersects circle $C_1$ at points $A$ and $B$, and intersects parabola $C_2$ at points $C$ and $D$.
(1) If $m=0$, as line $l$ rotates around point $M$, find the equation of the trajectory of the midpoint $R$ of segment $AB$;
(2) When $m=2$ and $\overrightarrow {AC}= \overrightarrow {DB}$, find the equation of line $l$.

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