
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 $y = a' \csc(b'x + c)$ for some positive constants $a'$, $b'$, and a phase shift $c$. Find $a'$.

[asy]
import TrigMacros;
size(500);

real f(real x)
{
	return 3*csc(2*x + pi/3);
}

draw(graph(f,-3*pi, -pi),red);
draw(graph(f,-pi, pi),red);
draw(graph(f, pi, 3*pi),red);
limits((-3*pi,-5),(3*pi,5),Crop);
draw((-pi,-5)--(-pi,5),dashed);
draw((pi,-5)--(pi,5),dashed);
trig_axes(-3*pi,3*pi,-5,5,pi/2,1);
layer();
rm_trig_labels(-3, 3, 1);

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

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