Resolving Symmetric Movement Paradox w/ Respect to Stationary Observer

[Kyojin]

I am trying to understand this apparent "paradoxes" but probably i am missing something important.
Imagine that accourding to stationary observer on Earth two twin cats are moving in the opposite directions with speed -v and v. When the two cats meet the stationary observer at the beginning O of his coordinate system K_{tx} their clocks are synchronized. Left cat has coordinate system K^{\prime}_{t^{\prime}x^{\prime}} right cat K^{\prime\prime}_{t^{\prime\prime}x^{\prime\prime}}.So at the center t_O=t^{\prime}_O=t^{\prime\prime}_O=0 and x_O=x^{\prime}_O=x^{\prime\prime}_O=0.
Now accourding to the stationary observer the two twin cats both travel T until they reach points x_{-S}=-vT and x_{S}=vT and then they go back and meet at the center.
Let \gamma_x=\frac{1}{\sqrt{1-\frac{x^2}{c^2}}}
Now let's take the viewpoint of the left cat.To find the time at witch the point x_{-s} reaches it we use the Lorentz transformation:
t^\prime_{-S}-t^\prime_{O}=\gamma_v(t_{-S}-t_{O}+\frac{vx_{-S}}{c^2}-\frac{vx_{-S}}{c^2})=\gamma_vT
Since the point x_{-S} stays stationary accourding to the unprimed frame.
Using the same calculation for the rigth cat the point x_S reaches it at time t^{\prime\prime}_S=\gamma_vT

Now accourding to the left cat the right cat is moving with speed w=\frac{2v}{1+\frac{v^2}{c^2}}. Now using the lorentz transformations again we can find that the right cat moves from from x_O to x_S accourding to the left cat for time t_S^\prime-t^\prime_O=t_S^\prime=\gamma_wt^{\prime\prime}_{S}=\gamma_w\gamma_vT.
And doing the same thing the left cat moves from x_{O} to x_{-S} accourding to the right cat for time t_{-S}^{\prime\prime}-t^{\prime\prime}_O=t_{-S}^{\prime\prime}=\gamma_w\gamma_vT.

Now if we do the same thing for the reverse direction at the end we will find the exactly same thing. Each cat thinks that the other is younger at the end of their path. But their situation is symmetric and they actualy did exactly the same thing. Shouldnt they age exactly the same at the end?
How can I resolve mathematicaly this disagreement on which cat is younger when from symmetry viewpoint they should be the same age?

I will appreciate any help. Thanks.

 

[ghwellsjr]

You've left out a whole lot of intermediate steps which now we have to repeat to find your error. Why don't you post all of your intermediate steps so that we don't have to? Maybe even when you do that, your error will be obvious.