Question about Sig-digs

I have no problem knowing how to find the right sig-digs when just doing one of: adding, subtracting, multiplying or dividing. But, I know doing them all at once in more complicated equations gets trickier.

For example, something like 110.0 + (500434.0 - 34.0)/0.37

This is my thought process for finding sig-digs when addition/subtraction is combined with multiplication/division. The answer may be correct by luck, but I know that this is not the right way to come up with the correct number of sig-digs.

110.0 + (500434.0 - 34.0)/0.37

= 110.0 + 500434.0/0.37

= 110.0 + 1.4*10^6 (I make 500434/0.37 equal 2 sig digs because the denominator only has 2 sig-digs.)

= 1.4001100*10^6 (Then I increase the sig-digs by 5 because I am adding 110.0.)

I know this process is wrong because my professor tried to explain to me that there is an intermediate step that I am missing. I can't grasp what it is though.