
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 that the line $ax+by+c=0$ always bisects the perimeter of the circle $C: x^2+y^2-2x+4y-4=0$ (where $C$ is the center of the circle), and let the line $l: (2a-b)x+(2b-c)y+(2c-a)=0$ pass through point $P(6,9)$. A perpendicular line from $P$ to $l$ meets at point $H$. Then, the range of the length of segment $CH$ is \_\_\_\_\_\_.

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