
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
If the points \( P_{1}(x_{1}, y_{1}), P_{2}(x_{2}, y_{2}), P_{3}(x_{3}, y_{3}) \) are on the curve \( xy=1 \), then the area of the triangle \( P_{1} P_{2} P_{3} \) is given by:

$$
t=\frac{\left(x_{1}-x_{2}\right)\left(x_{2}-x_{3}\right)\left(x_{3}-x_{1}\right)}{2 x_{1} x_{2} x_{3}}
$$

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