
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 parabola passing through points $A(2-3b, m)$ and $B(4b+c-1, m)$ is $y=-\frac{1}{2}x^{2}+bx-b^{2}+2c$, if the parabola intersects the $x$-axis, calculate the length of segment $AB$.

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