
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  $(1+\sin t)(1+\cos t)=5/4$  and \[ (1-\sin t)(1-\cos t)=\frac mn-\sqrt{k}, \] 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}
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