DGHV FHE, How is the maximum value of the rightmost term in Lemma A.1 calculated?

The proof of Lemma A.1 of the paper "FHE over the Integers" (Page 21-22) states that the absolute value of the rightmost term of

$c=p\cdot (kq_0+\sum_{i\in S}{q_i})+(m+2r+k\cdot2r_0+\sum_{i\in S}{r_i})$

is at most $(4\tau +3)2^{\rho}<\tau 2^{\rho+3}$.

We have $|k|\leq\tau$ and $|r_i|\leq 2^{\rho}$.

In addition, it is mentioned in the paper that $-2^{\rho'}<r<2^{\rho'}$ and $m\in\{0,1\}$.

My question is, how do we get $(4\tau +3)2^{\rho}$?

I calculated the maximum value for $(m+2r+k\cdot2r_0+\sum_{i\in S}{r_i})$ as $(1+2^{\rho'+1}+(4\tau-2)2^{\rho})$ (I take $k=\tau -1$ and $S=\{1,\ldots,\tau\}$ as upper bounds).

I couldn't get $(4\tau +3)2^{\rho}$ even if I put the suggested value of $\rho'$ (which is $2\rho$). I can't figure out where I'm wrong.