Keep in mind that in order to win a block reward, you need to be in the list of N2 miners of other miners.
The first N2 solutions received by one miner could be different from the first N2 received by another. Do you have a way to achieve consensus on who is in the list? If there are different lists, then the network will split when miners don't agree on who is in the list.
No. But we don't need it. If there are different lists the network with go with the longest chain like in BTC.
Again like in Bitcoin, we take the longest chain. No difference with Bitcoin. The honest and fastest miner will always win in the long run.
If you don't accept a block by someone not in your list, then you can't follow the actual longest chain. If you allow a block from a miner not in your list then
every miner can include themselves in their own list (as soon as they complete round 1), which you must accept in order to resolve a split.
Please Read instructions 1) and 5) below.
===========TRPoW miners will execute the following instructions============================
1- If a miner receive $N_2$ $nonce1$ or more broadcast from other miners before finding its own $nonce1$, it will broadcast the list of the first $N_2$ winners and drop the current block competition and move to the next block. In this case it failed the first round and the block reward.
2- If it finds its $nonce1$ before receiving $N_2 - 1$ solutions from other miners, it will broadcast its $nonce1$ with the list of all the miners that found the solution before it including itself. it will continue listening the network to build the full list of the first $N_2$ winners. And at the same time move to the next round competition. In this case it succeed the first round.
3- If it didn't find its $nonce1$ yet and did not receive yet the full list of $N_2$ winners, it will continue solving for its first round $nonce1$ and listening to the network broadcasting to build its list of $N_2$ winners. In this case it is still running the first round.
4- If it finds its $nonce2$ before receiving any valid $nonce2$, it will broadcast its $nonce2$ and update the ledger and register the block reward for itself. And move to the next block. In this case it won the current block reward.
5- If it receive an $nonce2$, it will verify the solution and verify that the miner is in its list of winners of the first round. If all is fine, it will move to the next block competition having failed the current block competition. If the miner is not in its list of winners of the first round, it will ignore what it received and continue solving for its own $nonce1$ or / and $nonce2$ and listening to the network.