
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 three congruent triangles $\triangle ABP$, $\triangle ACP$, and $\triangle ADP$, such that $AB = AC = AD = BP = CP = DP,$ as shown in the diagram. If $\angle BAP = 15^\circ,$ determine the measure of $\angle BDP$. 
[asy]
pair pA, pB, pC, pD, pP;
pA = (0, 0);
pB = pA + dir(240);
pC = pA + dir(260);
pD = pA + dir(280);
pP = pA + 0.5*dir(300); // Middle point
draw(pA--pB--pP--pA);
draw(pA--pC--pP--pA);
draw(pA--pD--pP);
label("$A$", pA, N);
label("$B$", pB, SW);
label("$C$", pC, S);
label("$D$", pD, SE);
label("$P$", pP, E);
[/asy]

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