I am studying for an exam on Special Relativity, and I made a toy problem to practice, but I ended up confusing myself. I know something is wrong with my intuition, but can someone please point where I am making the error? Here's the problem:

Consider a rocket traveling with along the x-axis with constant velocity 0.8c. At some time $t'_0=0s$, the rockets starts a clock and emits a photon. At $t'_1=0.6s$ he stops the clock. What is the separation between the rocket and the photon for the rockets reference frames, and for a reference frame at rest?

Here's my work: okay so from the reference frames of the rocket, the rocket appears to be at rest, so the separation would be the distance the photon travels: $ 0.6 \cdot 3 \cdot 10^8 = 1.8 \cdot 10^8m$

From a stationary rest frame we find

$ \gamma = (1-\frac{v^2}{c^2})^\frac{-1}{2}= 1.\overline{6} $

$t_1 = t_1' \cdot γ = 1s$

The rocket is moving at 0.8c so it travels $ 0.8 \cdot 3 \cdot 10^8 = 2.4 \cdot 10^8 $

The photon travels $ 3 \cdot 10^8 m $

So the separation would be $ (3-2.4) \cdot 10^8 = 0.6 \cdot 10^8m. $ If we apply length contraction we get $0.6 \cdot 10^8 \cdot \gamma = 1 \cdot 10^8$ , but this does not agree with the previous calculation of $ 1.8 \cdot 10^8m $ separation.

Can someone help me understand what I'm doing wrong?

(Apologies if this has terrible formatting, I'm writing from a mobile device!)