
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
Below is the graph of \( y = a \csc bx \) for some positive constants \( a \) and \( b \). Assume \( a = 3 \). The vertical dashed lines are located at \( x = -2\pi \) and \( x = 2\pi \). Find \( b \).

\[ 
\text{[asy]}
import TrigMacros;

size(300);
real g(real x)
{
    return 3*csc(x/2);
}

draw(graph(g, -4\pi + 0.01, -2\pi - 0.01), red);
draw(graph(g, -2\pi + 0.01, 0.01), red);
draw(graph(g, 0.01, 2\pi - 0.01), red);
draw(graph(g, 2\pi + 0.01, 4\pi - 0.01), red);
limits((-4\pi, -5), (4\pi, 5), Crop);
draw((-2\pi, -5) -- (-2\pi, 5), dashed);
draw((2\pi, -5) -- (2\pi, 5), dashed);
trig_axes(-4\pi, 4\pi, -5, 5, \pi/2, 1);
layer();
rm_trig_labels(-11, 11, 2);

label("$1$", (0, 1), E);
label("$3$", (0, 3), E);
label("$-1$", (0, -1), E);
label("$-3$", (0, -3), E);
\text{[/asy]}
\]

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