Consider the following question: If two nodes collide what do you get? First of all it can not be a strict $A_2$ node, because the delta invariant of that is $1$. So it has to be more singular than an $A_2$ node. It can be an $A_3$ node because the delta invariant of that is $2$.

Is there any simple argument to show that if an $A_2$ node and an $A_1$ node collide, then we can not get a strict $A_3$ node? The delta invariant doesn't help. Is there some other invariant that can answer this question? Note that I am NOT asking what do we actually get when an $A_2$ node and an $A_1$ node collide. I merely want to show that we can not get a strict $A_3$ node.