
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 three real numbers $a$, $b$, and $c$ form an arithmetic sequence. A perpendicular line to $ax+by+c=0$ is drawn from point P $(-3, 2)$ and meets it at point M. Given the point N $(2, 3)$, find the range of lengths for segment $MN$.

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