
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
Suppose there are two congruent triangles $\triangle ABC$ and $\triangle ACD$ with $AB = AC = AD$. Let the angle $\angle BAC = 30^\circ$ in the diagram. Find the measure of $\angle BDC$. The configuration is shown below:
[asy]
pair pA, pB, pC, pD;
pA = (0, 0);
pB = pA + dir(225);
pC = pA + dir(255);
pD = pA + dir(285);
draw(pA--pB--pC--pA);
draw(pA--pC--pD--pA);
draw(pB--pD,red);
label("$A$", pA, N);
label("$B$", pB, SW);
label("$C$", pC, S);
label("$D$", pD, SE);
[/asy]

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