
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
Two congruent 30-60-90 triangles are positioned such that they partly overlap, and a portion of their hypotenuses coincide. If the hypotenuse of each triangle is 10 units instead of 12, determine the area common to both triangles when the overlapping section of their hypotenuses is 5 units.
**A)** \( \frac{15\sqrt{3}}{4} \)
**B)** \( \frac{25\sqrt{3}}{4} \)
**C)** \( \frac{20\sqrt{3}}{3} \)
**D)** \( \frac{10\sqrt{3}}{2} \)
**E)** \( \frac{30\sqrt{3}}{6} \)

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