
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
A  $39$ -tuple of real numbers  $(x_1,x_2,\ldots x_{39})$  satisfies
\[2\sum_{i=1}^{39} \sin(x_i) = \sum_{i=1}^{39} \cos(x_i) = -34.\] 
The ratio between the maximum of  $\cos(x_1)$  and the maximum of  $\sin(x_1)$  over all tuples  $(x_1,x_2,\ldots x_{39})$  satisfying the condition is  $\tfrac ab$  for coprime positive integers  $a$ ,  $b$  (these maxima aren't necessarily achieved using the same tuple of real numbers). Find  $a + b$ .

*Proposed by Evan Chang*

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