The evil sorceress Morgana lives in a square-shaped palace divided into a \(101 \times 101\) grid of rooms, each initially at a temperature of \(20^\circ\)C. Merlin attempts to freeze Morgana by casting a spell that permanently sets the central cell's temperature to \(0^\circ\)C. At each subsequent nanosecond, the following steps occur in order: 1. For every cell except the central one, the new temperature is computed as the arithmetic mean of the temperatures of its adjacent cells (those sharing a side). 2. All new temperatures (except the central cell) are updated simultaneously to the calculated values. Morgana can freely move between rooms but will freeze if all room temperatures drop below \(10^{-2025}\) degrees. The ice spell lasts for \(10^{75}\) nanoseconds, after which temperatures revert to their initial values. Will Merlin succeed in freezing Morgana?