Markov Chains & Long-Run Behavior · Superchallenge problem

A token moves on the four corners of a square. Each step it stays put with probability $\tfrac12$, and otherwise it moves to one of the two neighbouring corners, each with probability $\tfrac14$. Let $M$ be the $4\times4$ transition matrix. What is the smallest positive integer $n$ such that every entry of $M^n$ is within $0.01$ of $\tfrac14$?