Given:
\[ |2a - 1| = 3[a] + 2\{a\} \]
Rewrite \(|2a - 1|\) in two forms depending on the value of \(a\):
In this case:
\[ 2a - 1 = [a] + 2a \]
Since \([a] = -1\), we find that \(a \in [-1, 0)\), which is a contradiction because \(a > \frac{1}{2}\). Therefore, this case is rejected.
Case 2: \(a < \frac{1}{2}\) In this case:
\[ -2a + 1 = [a] + 2a \]
Let \(a = I + f\) where \(I\) is the integer part and \(f\) is the fractional part, so \([a] = 0\) and \(\{a\} = f\).
Then we have:
\[ -2(I + f) + 1 = I + 2f \]
Substituting \(I = 0\), we get:
\[ 1 = 2f \implies f = \frac{1}{4} \]
Thus, \(a = \frac{1}{4}\).
Now, calculating \(72 \sum_{a \in S} a\): \[ 72 \times \frac{1}{4} = 18 \]
Let \( A = \{-3, -2, -1, 0, 1, 2, 3\} \). A relation \( R \) is defined such that \( xRy \) if \( y = \max(x, 1) \). The number of elements required to make it reflexive is \( l \), the number of elements required to make it symmetric is \( m \), and the number of elements in the relation \( R \) is \( n \). Then the value of \( l + m + n \) is equal to: