Freeze the seat and heat the block and tap in I would replace the seat if it fell out something is wrong
Valve in head, or valve in block?
May need a machinest to fit an oversize seat. Which means the engine has to scattered if it's a valve in block. Which means it might be cheaper to install a shortblock or entire engine. If It's a valve in head, then just have a machinest fix it with an oversize seat.
If you think a new seat will fix the deal, I believe they use a super-duper hi-heat loc-tite to retain valve seats. But a new seat will likely need to have the valve angle ground.
You could try to reinstall the original seat, but if it's not perfectly back in place, or not concentric with the valve itself(need reground), will cause all sorts of issues including loss of compression, backfiring, may eventualy burn the valve.
That's if the seat stays in place this time.
I would get a new seat just to be safe.
I used to to this exact repair all the time. All you do is to put the seat back in (again I recommend that you get a new one) and then take a center punch and work your way around the seat punching the aluminum thus causing the seat to be pinched in against the block. First you do 4 points around the seat to make it seat evenly, then punch in the rest of the way around the seat. It's very much like staking a gas carrier key.
Install the seat into the block and then reassemble the valve spring. You want there to be some pressure on the seat from the valve to keep it in completely while you punch the aluminum.
Once you are done, even if you re-use the seat, you must lap it in to reseat it. I would do both valves and then also relap the other set of valves on the other head. Basically you are doing a valve job to the engine.
This will take some special tools but they are all pretty inexpensive.
I've never seen a repair like this, done properly, fail.