
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
Given that $(1+\sin t)(1+\cos t)=\frac{9}{4}$ and
$(1-\sin t)(1-\cos t)=\frac{m}{n}-\sqrt{k},$
where $k, m,$ and $n$ are positive integers with $m$ and $n$ relatively prime, find $k+m+n.$

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