Stopping at $r = \lfloor\pi/(4t)\rfloor$, $\omega$ is observed with error at most $1/N$

Let $f\colon\{0,\dots,N-1\}\to\{0,1\}$ have exactly one index $\omega$ with $f(\omega) = 1$, let $t = \arcsin(1/\sqrt N)$ and $r = \lfloor \pi/(4t)\rfloor$. After $r$ Grover iterations, measuring the state yields $\omega$ with probability

i.e. with error at most $1/N$. This is a quantitative form of the source's claim that stopping at $r = \pi/4t \approx \pi\sqrt N/4$ gives the correct answer with error $O(1/N)$.

Formalization Note The source's $r = \pi/4t$ is in general not an integer; the mission rounds it down, and replaces "$O(1/N)$" by the explicit bound $1/N$.