
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
In the Cartesian coordinate system, $O$ is the origin, and $P_1(x_1, y_1)$, $P_2(x_2, y_2)$ are two points in the first quadrant. If $1, x_1, x_2, 4$ form an arithmetic sequence, and $1, y_1, y_2, 8$ form a geometric sequence, find the area of triangle $\triangle OP_1P_2$.

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