
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
A square and an isosceles triangle of equal height are side-by-side on the $x$-axis, with both bases aligned on the axis. Adjacent to the triangle is a parallelogram of the same height as the triangle and a base of 8 units on the $x$-axis. The lower right vertex of the square and the lower left vertex of the triangle are at $(12, 0)$. The side of the square and the base of the triangle on the $x$-axis each equal $12$ units. A segment is drawn from the top left vertex of the square to the farthest vertex of the parallelogram. Calculate the area of the shaded region formed.

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