
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 $y = a \cos (bx + c) + d$ for some constants $a,$ $b,$ $c,$ and $d.$  Find $b$. The graph indicates that the function has a period that covers two periods in an interval of $3\pi$, determining the period observed in the graph to be $\frac{3\pi}{2}$.

[asy]import TrigMacros;

size(400);

real f(real x)
{
	return 2*cos(3*x/2 + pi) + 1;
}

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