
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
In the diagram line segments  $AB$  and  $CD$  are of length 1 while angles  $ABC$  and  $CBD$  are  $90^\circ$  and  $30^\circ$  respectively.  Find  $AC$ .

[asy]
import geometry;
import graph;

unitsize(1.5 cm);

pair A, B, C, D;

B = (0,0);
D = (3,0);
A = 2*dir(120);
C = extension(B,dir(30),A,D);

draw(A--B--D--cycle);
draw(B--C);
draw(arc(B,0.5,0,30));

label(" $A$ ", A, NW);
label(" $B$ ", B, SW);
label(" $C$ ", C, NE);
label(" $D$ ", D, SE);
label(" $30^\circ$ ", (0.8,0.2));
label(" $90^\circ$ ", (0.1,0.5));

perpendicular(B,NE,C-B);
[/asy]

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