
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
In the Cartesian coordinate system $xOy$, given circle $C_1: x^2+y^2-6x+4y+9=0$, and circle $C_2: (x+m)^2+(y+m+5)^2=2m^2+8m+10$ ($m \in \mathbb{R}$, and $m \neq -3$).

(I) If $m=5$, find the number of intersection points between circle $C_1$ and circle $C_2$.

(II) Let $P$ be a point on the coordinate axes, satisfying: tangents to circles $C_1$ and $C_2$ are drawn from point $P$, with the points of tangency being $T_1$ and $T_2$ respectively, such that $PT_1=PT_2$. Find the coordinates of all such points $P$.

(III) If a line $l$ with slope $k$ bisects circle $C_1$ and always intersects circle $C_2$, find the range of the slope $k$ of line $l$.

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