I power my system on and the screen doesn't start up I've checked all my cords and they...

Hi, I'm Luis a Xbox user just like you. Thanks for reaching out us here.

In order to hard reset you need to:

Turn off your console by pressing the Xbox button  on the front of the console for about 10 seconds, until it shuts down completely.
Unplug the console's power cable. Wait 10 seconds.
Important Be sure to wait at least 10 seconds. This step resets the power supply.
Plug the console power cable back in.
Press the Xbox button  on your console to turn it on.

If it doesn't work then I would suggest you to have your device repaired

I'm having this same problem. My Xbox one keeps restarting and I've checked all of my power cables and they all seem to be working fine. It seems to be related to the wifi as well but I don't have a way to get an Ethernet cord from my router to my Xbox.
Has anyone managed to find a solution that doesn't include wiring the Xbox?

Try performing a hard reset. Hold the power button for 10 seconds while the system is on. The system will turn off. Disconnect the power cord for 3 minutes as well. After several minutes plug it back in, then hit the power button again. It will start and
display the green start-up screen. All your data is preserved, but cache is cleared and some settings are reverted and log-in or game start issues should be resolved. To be thorough, I would also try signing out of your account then signing back in.

are you asking if you can use the old power brick for a newer console? if so i have an adapter that turns the old model power cord into a slim capable power cord by changing the end that plugs into the system itself.. and i've had no issues with that, i
use it all the time to move my Slim to the location of my OG xbox for the sake of rock band parties. I hope that helps!

All I had to do was a Hard Restart / Full Reboot of my Xbox.

To do that, hold the power button on the xbox for 10 seconds while the system is on. The system will turn off. When you hit the button again, it will start and display the green start-up screen.