right before $O_{2}$ receives its kick of energy is when the phase of this clock is $2\pi^{-}$.

Then we have $$%
\begin{cases}
\psi_{2}^{4} & =2\pi^{-}\\
\psi_{1}^{4} & =\psi_{1}^{3}+2\pi-\psi_{2}^{3}\\
& \simeq2\pi-{\psi}_{3}^{0}-2P\left(  {\psi}_{3}^{0}\right)  +2\pi\\
& -\left(  2\pi+{\psi}_{2}^{0}-{\psi}_{3}^{0}+P\left(  {\psi}_{2}^{0}\right)
-P\left(  {\psi}_{3}^{0}\right)  +P\left(  {\psi}_{2}^{0}-{\psi}_{3}%
^{0}\right)  \right) \\
& =2\pi-{\psi}_{2}^{0}-P\left(  {\psi}_{2}^{0}\right)  -P\left(  {\psi}%
_{3}^{0}\right)  -P\left(  {\psi}_{2}^{0}-{\psi}_{3}^{0}\right) \\
\psi_{3}^{4} & =\psi_{3}^{3}+2\pi-\psi_{2}^{3}\\
& \simeq2\pi+{2\pi}-\left(  2\pi+{\psi}_{2}^{0}-{\psi}_{3}^{0}+P\left(  {\psi
}_{2}^{0}\right)  -P\left(  {\psi}_{3}^{0}\right)  +P\left(  {\psi}_{2}%
^{0}-{\psi}_{3}^{0}\right)  \right) \\
& \simeq2\pi-{\psi}_{2}^{0}+{\psi}_{3}^{0}-P\left(  {\psi}_{2}^{0}\right)
+P\left(  {\psi}_{3}^{0}\right)  -P\left(  {\psi}_{2}^{0}-{\psi}_{3}%
^{0}\right)  .
\end{cases}$$

**STEP 5**: third impact. Clock $O_{2}$ receives its internal energy kick. It reaches the position $2\pi$.

Then we have $$%
\begin{cases}
\psi_{2}^{5} & ={2\pi}\\
\psi_{3}^{5} & =\psi_{3}^{4} +P(\psi_{3}^{4})\\
& \simeq2\pi-{\psi}_{2}^{0}+{\psi}_{3}^{0}-P\left(  {\psi}_{2}^{0}\right)
+P\left(  {\psi}_{3}^{0}\right)  -P\left(  {\psi}_{2}^{0}-{\psi}_{3}%
^{0}\right) \\
& +P(2\pi-{\psi}_{2}^{0}+{\psi}_{3}^{0}-P\left(  {\psi}_{2}^{0}\right)
+P\left(  {\psi}_{3}^{0}\right)  -P\left(  {\psi}_{2}^{0}-{\psi}_{3}%
^{0}\right)  )\\
& \simeq2\pi-{\psi}_{2}^{0}+{\psi}_{3}^{0}-P\left(  {\psi}_{2}^{0}\right)
+P\left(  {\psi}_{3}^{0}\right)  -P\left(  {\psi}_{2}^{0}-{\psi}_{3}%
^{0}\right)  -P({\psi}_{2}^{0}-{\psi}_{3}^{0})\\
& =2\pi-{\psi}_{2}^{0}+{\psi}_{3}^{0}-P\left(  {\psi}_{2}^{0}\right)
+P\left(  {\psi}_{3}^{0}\right)  -2P\left(  {\psi}_{2}^{0}-{\psi}_{3}%
^{0}\right) \\
\psi_{1}^{5} & =\psi_{1}^{4}+P\left(  \psi_{1}^{4} \right) \\
& \simeq2\pi-{\psi}_{2}^{0}-P\left(  {\psi}_{2}^{0}\right)  -P\left(  {\psi
}_{3}^{0}\right)  -P\left(  {\psi}_{2}^{0}-{\psi}_{3}^{0}\right)  +\\
& P\left(  2\pi-{\psi}_{2}^{0}-P\left(  {\psi}_{2}^{0}\right)  -P\left(
{\psi}_{3}^{0}\right)  -P\left(  {\psi}_{2}^{0}-{\psi}_{3}^{0}\right)  \right)
\\
& \simeq2\pi-{\psi}_{2}^{0}-P\left(  {\psi}_{2}^{0}\right)  -P\left(  {\psi
}_{3}^{0}\right)  -P\left(  {\psi}_{2}^{0}-{\psi}_{3}^{0}\right)  -P({\psi
}_{2}^{0})\\
& =2\pi-{\psi}_{2}^{0}-2P\left(  {\psi}_{2}^{0}\right)  -P\left(  {\psi}%
_{3}^{0}\right)  -P\left(  {\psi}_{2}^{0}-{\psi}_{3}^{0}\right)  .
\end{cases}$$

**STEP 6 (the final)**: third natural time shift. The next clock to arrive at 