Perl, 65 58 bytes
map{map{$b^=2*y,/\\L,,;$a+=y,/\\V,,||$b}split//}<>;print$a
- Toggle $b between 0 and 2 upon seeing / \ or L.
- Add 1 to $a upon seeing / \ or V.
- Add $b to $a upon seeing anything else.
map{map{$b^=2*y,/\\L,,;$a+=y,/\\V,,||$b}split//}<>;print$a