Brute Force?

Algebra Level 3

\[ \large \left \lceil \lfloor \pi \rfloor + \left\lfloor \pi + \dfrac{1}{6}\right \rfloor + \left \lfloor \pi + \dfrac{2}{6} \right \rfloor + \left \lfloor \pi + \dfrac{3}{6} \right \rfloor + \left \lfloor \pi + \dfrac{4}{6}\right \rfloor +\left \lfloor \pi + \dfrac{5}{6}\right \rfloor \right \rceil + 1 = \, ? \]

\[\] Notations: \( \lfloor \cdot \rfloor \) denotes the floor function and \( \lceil \cdot \rceil \) denotes the ceiling function.