In the setting of $25$-strand braids, let $\sigma_i$ for $i=1,\dots,24$ denote the positive braid generator in which the $i$-th strand crosses over the $(i+1)$-st strand.

Consider the braid diagram $\beta$ obtained by stacking the standard diagrams of the generators $\sigma_{24}\sigma_{23}\cdots\sigma_1 \sigma_2\sigma_3\cdots\sigma_{24}$ from bottom to top.

Construct an admissible system of $25$ curves as follows. Process the crossings of $\beta$ from top to bottom, and at each crossing push the incoming understrand (the lower branch) along the overstrand (the upper branch) to the outgoing endpoint of the overstrand (the upper endpoint).

This procedure yields a family of $25$ pairwise disjoint simple curves in the upper half-plane connecting the bottom points $(i,0)$ to the top points $(j,1)$ for $i,j=1,\dots,25$, and hence determines a permutation $\pi\in S_{25}$ defined by $\pi(i)=j$.

It is given that $\pi$ consists only of fixed points and disjoint adjacent transpositions. Compute the number of fixed points of $\pi$.