If \(A = \lceil \sqrt{3} \rceil + \lceil \sqrt{4} \rceil + \lceil \sqrt{5} \rceil +\cdots + \lceil \sqrt{83} \rceil \), find the value of \( \dfrac13\left \lceil \sqrt A \right \rceil \).

**Notation**: \( \lceil \cdot \rceil \) denotes the ceiling function.

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