
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
Derive a formula for computing the area of an n-sided polygon $A_{1}A_{2}\ldots A_{n}$, given that its vertices are numbered in the same order as in the previous problem and are specified by their coordinates: $A_{1}\left(x_{1}, y_{1}\right), A_{2}\left(x_{2}, y_{2}\right), \ldots, A_{n}\left(x_{n}, y_{n}\right)$.

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