
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
There is a positive integer s such that there are s solutions to the equation $64sin^2(2x)+tan^2(x)+cot^2(x)=46$  in the interval  $(0,\frac{\pi}{2})$  all of the form  $\frac{m_k}{n_k}\pi$ where  $m_k$  and  $n_k$  are relatively prime positive integers, for  $k = 1, 2, 3, . . . , s$ . Find  $(m_1 + n_1) + (m_2 + n_2) + (m_3 + n_3) + · · · + (m_s + n_s)$ .

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