
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
The area of the plane region represented by the system of inequalities 
\[
\begin{cases}
y \leqslant -x + 2, \\
y \leqslant x - 1, \\
y \geqslant 0
\end{cases}
\]
is given by the formula $\dfrac{1}{2}\left|x_{2}x_{3} - x_{1}x_{3} + x_{1}x_{2}\right]$, where $(x_1, x_2, x_3)$ are the x-coordinates of the intersection points of the lines. Calculate this area.

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