
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
Let \(ABCD\) be a rhombus, with \(\angle BAD > \angle ABC\), and let \(P\) and \(Q\) be points on sides \(AB\) and \(AD\), respectively, such that the triangle \(PCQ\) is equilateral with side lengths equal to the side length of the rhombus. Find the measures of the angles of the rhombus.

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