What does Putin think about the West arming Ukraine?

The US is pouring ammunition and small arms into Ukraine by the multi-tons worth, and the UK is contemplating sending more than the 2,000 anti-tank missile launchers it has already flown to Kyiv by C-17 transport aircraft. How will this be viewed in the Kremlin? And more specifically, will this give Putin the extra bit of motivation he needs to order his 100,000 troops over the border and down into the whole of eastern Ukraine and beyond? Or will it act as a deterrent? A better-armed Ukraine will mean a longer and bloodier war. The one thing Putin DOES know is that Joe Biden and the rest of Nato are never going to join in the war. Whatever Nato does it won't be fighting alongside the Ukrainian army. First, because Ukraine is not a member of Nato and therefore there is no obligation to do so, and second, because the last thing Biden wants is another war on his watch. Afghanistan was a disaster for the US and the way US military involvement was brought to an abrupt end was nothing short of shameful. Afghanistan is now in an economic, human rights black hole and malnutrition is everywhere. So Biden will never take on the Russians in Ukraine. Arming Kyiv is the only way to get back at Putin, plus launching severe economic sanctions in the event of an invasion and reinforcing eastern Europe. So with all this ammo arriving for the Ukrainian army, could it just be a step too far in Putin's mind, making him even more determined to teach Ukraine a lesson? I fear that may be the case. He will think to himself, "if the West is arming Ukraine now, it won't be long before they hand them membership of Nato, so it's time to go go go". In other words, if he invades he will at least have a chance of taking control of what happens in Ukraine in the future, whereas if he pulls his troops back he will no longer have a big stick to wave to prevent Ukraine from rushing into alliance membership, and before he knows it there will be American troops settling into a barracks outside Kyiv and wholesale Nato execises taking place the other side of the Russian border. So, yes, I believe the US/UK arming of Ukraine might just stir Putin to go for broke and order an invasion.