ISL 2023 C3

Let $n$ be a positive integer. We arrange $1+2+\cdots+n$ circles in a triangle with $n$ rows, such that the $i^{\text {th }}$ row contains exactly $i$ circles. The following figure shows the case $n=6$. In this triangle, a ninja-path is a sequence of circles obtained by repeatedly going from a circle to one of the two circles directly below it. In terms of $n$, find the largest value of $k$ such that if one circle from every row is coloured red, we can always find a ninja-path in which at least $k$ of the circles are red. (Netherlands)