What's the redshift of a return signal in a flat universe between two observers?

I’m trying to determine the redshift observed for a light signal between 2 observers (let’s call them observers one and two) in a flat matter-dominated universe. Observer one sends a signal ($t_1$) at light speed to observer 2 and she records it as a z of 5 ($t_2$). Observer two then sends a signal directly back. At what redshift does observer one see this incoming signal ($t_3$)?

We know the scale factor for a matter-dominated universe is as follows:
$$
frac{a}{a_0}=frac{{t^{2/3}}}{t_0}
$$

and $a_0$ and $t_0$ are $1$ for the present time

Additionally, the comoving coordinate will stay the same for both time intervals by definintion and can be described as follows:
$$
r=cintfrac{dt}{a(t)}
$$

Using the relation between the scale factor and z, we can find the following relationship:
$$
a(t_2) = 6cdot a(t_1)
$$

Describe the comoving coordinates for the time intervals and set them equal to each other:
$$
int_{t_1}^{t_2}frac{dt}{{t}^{frac{2}{3}}} = int_{t_2}^{t_3}frac{dt}{{t}^{frac{2}{3}}}
$$

After integration, substitute the relation found above in terms of t to get the ratio between $t_1$ and $t_3$:
$$
t_2 = 6^{frac{3}{2}}cdot t_1
$$
…to get:
$$
2(14.7cdot t_1)^{1/3}-t_1^{1/3} = t_3^{1/3}
$$

Lastly, now that we have a relationship between the initial signal emitted ($t_1$) and the final signal received ($t_3$), we can determine $z$:
$$
frac{a(t_3)}{a(t_1)}=1+z=frac{t_3^{frac{2}{3}}}{left [ frac{t_3}{59} right ]} therefore zapprox 14
$$

Above is the work I did but I’m not confident my answer is correct although I don’t see anything glaring wrong with what I did. I expect the redshift to be greater which is consistent with the redshift calculated, but I don’t have much greater intuition that this. Perhaps there is a more elegant way to solve it. I’d appreciate if someone could let me know if the solution seems sound.

Thanks