Putin knows that Biden is an incredibly weak president. China's Xi knows this also. Putin and Xi met a few days ago for discussions. It doesn't take a genius to guess what they were talking about. They want to coordinate, to double team the US internationally. Putin is interested in Ukraine, while Xi wants to advance in the South China Sea and with Taiwan.
I seriously doubt that either of them has an exact end game for this. They just keep pushing their goals as far as they can. And, with coordination, they can get that extra inch of progress towards their goals.
Maybe others in the US government can fill in for Biden? US General Milley told the Chinese that he would tell the Chinese first, before launching any nuclear strike against them. America's top leaders are that stupid? They sure are!