MMO2025 Round 4

Let $\varphi (n)$ denote the number of positive integers less than or equal to a given integer $n$ that are relatively prime to $n$. For example, $\varphi (6)=|\{1,5\}|=2$ and $\varphi (1)=|\{1\}|=1$. Let $a,b,c,d$ be nonnegative integers such that $\varphi (2^a(2b+1))=2^c(2d+1)$. If $b\ge 1$ then show that (1) $a\le c$ and, (2) $b\ge 2d+1$.