
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
\((1-\sin t)(1-\cos t) = \frac{p}{q} - \sqrt{r},\)
where \( r, p, \) and \( q \) are positive integers with \( p \) and \( q \) relatively prime, find \( r+p+q. \)

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