7
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Python 3, 376 bytes
Basically a golfed version of the reference solution @DLosc gave
import re
k=" "
def p(e,*t,l=0):
for n in e:
if")"==n:break
if")">n:z=p(e);l=max(l,z[0]+1);t+=z,
else:l=max(l,1);t+=n,
return[l,*t]
t=p(iter(re.findall("[^() ]+|\S",input())[1:]))
def f(t,d=k):
if[*t]==t...