
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 $y = a \sin (bx + c) + d$ for some positive constants $a,$ $b,$ $c,$ and $d.$ The graph completes exactly 5 full oscillations as $x$ varies from $0$ to $2\pi$. Find the value of $b$.

[asy]
import TrigMacros;

size(400);

real f(real x)
{
	return 2*sin(5*x + pi) + 1;
}

draw(graph(f,0,2*pi,n=700,join=operator ..),red);
trig_axes(0,2*pi,-3,4,pi/5,1);
layer();
rm_trig_labels(-4,4, 2);

label("$1$", (0,1), E);
label("$2$", (0,2), E);
label("$3$", (0,3), E);
label("$-1$", (0,-1), E);
label("$-2$", (0,-2), E);
label("$-3$", (0,-3), E);
[/asy]

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