
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
The diagram shows a semicircle with diameter \(PQ\) inscribed in a rhombus \(ABCD\). The rhombus is tangent to the arc of the semicircle in two places. Points \(P\) and \(Q\) lie on sides \(BC\) and \(CD\) of the rhombus respectively. The line of symmetry of the semicircle is coincident with the diagonal \(AC\) of the rhombus. It is given that \(\angle CBA = 60^{\circ}\). The semicircle has radius 10. The area of the rhombus can be written in the form \(a\sqrt{b}\) where \(a\) and \(b\) are integers and \(b\) is prime. What is the value of \(ab + a + b?\)

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