A Partitioned Equilateral Triangle

Geometry Level 3

Above shows an equilateral triangle \(ABC\) with side length 1. Given that \( M, N\) and \(Q\) are points on the sides \(AB, BC\) and \(AC\) such that the lines \(AN, BQ\) and \(CM\) divide the triangle into 4 triangles and 3 quadrilaterals.

We color the triangles in two colors (orange and green) in such way that any two triangle with a common vertex are colored with different colors.

If the area colored in green is equal to that colored in orange, find the value of \( AM+BN+CQ \).

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