
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 right triangle are placed such that their bases are on the $x$-axis and they share a vertex at $(10, 0)$. The dimensions of the square are such that each side measures $12$ units. The right triangle has its right angle at $(10, 0)$, with the other two vertices being $(22, 0)$ and $(10, 18)$. A line segment is drawn from the top left vertex of the square to the hypotenuse of the triangle, intersecting at a certain point. Determine the area of the shaded region formed by this intersection.

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