Pi and triangles? No!

Geometry Level 4

A pentagon has respective side lengths \(a\), \(a \sqrt 3\), \(a\), \(a\), and \(a\), and angles \(90^\circ\), \(90^\circ\), \(120^\circ\), \(120^\circ\), and \(120^\circ\), both corresponding. This pentagon can be split into an equilateral triangle and two other triangles.

Find the smallest positive integer of \(a\) which satisfies the condition the product of the areas of the three triangles divided by their sum is greater than \( 100000 \pi\).

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