
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  $p(x)$  be the monic cubic polynomial with roots  $\sin^2(1^{\circ})$ ,  $\sin^2(3^{\circ})$ , and  $\sin^2(9^{\circ})$ . Suppose that  $p\left(\frac{1}{4}\right)=\frac{\sin(a^{\circ})}{n\sin(b^{\circ})}$ , where  $0 <a,b \le 90$  and  $a,b,n$  are positive integers.  What is  $a+b+n$ ?

*Proposed by Andrew Yuan*

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