What prevents a node from receiving a solved solution and claiming it as its own work? During the relay process a node finds the solution and forwards that solved block to all other nodes in the network to validate that it has indeed solved the problem. But then what if another node decides to rebroadcast it again and claim that he was the one who found the answer first.
Do you modify the block or do you not modify the block?

If you don't modify the block, it still pays the mining reward to the same account. So you've just done what everyone wants you to do.

If you do modify the block, it won't be valid anymore until and unless you mine it. That leaves you two choices:

1) You can ignore this block and mine some other block. But then someone else who mines on top of this block will have a longer chain than you and you won't get any reward. So why would you do this?

2) You can mine on top of this block. Well, that will just help ensure the person who mined that block gets their reward by producing an even longer chain that includes their block.