
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 \( y = a \sec (bx + c) \) for some positive constants \( a \), \( b \), and \( c \). Find the value of \( a \).

[asy]
import TrigMacros;

size(500);

real g(real x)
{
    return 3*sec(x/3 + pi/4);
}

draw(graph(g,-6*pi + 0.01, -3*pi - 0.01),red);
draw(graph(g,-3*pi + 0.01,-0.01),red);
draw(graph(g,0.01,3*pi - 0.01),red);
draw(graph(g,3*pi + 0.01,6*pi - 0.01),red);
limits((-6*pi,-5),(6*pi,5),Crop);
draw((-3*pi,-5)--(-3*pi,5),dashed);
draw((3*pi,-5)--(3*pi,5),dashed);
trig_axes(-6*pi,6*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);
[/asy]

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