# Matching a list of sequences with common form

I am trying to find a pattern that matches a list of sequences that share a common form. The sequence form is: 1 followed by zero or more nonzero integers, followed by 0. An example list is:

{1,3,2,0,1,0,1,2,0,1,1,4,0}


The closest thing I've struck on so far is this:

MatchQ[{1, 2, 0, 1, 2, 0}, {PatternSequence[1, s___ /; FreeQ[{s}, 0], 0] ..}]


which yields True in this case because the sequences are both the same. Is there a way to get the s___ to apply uniquely to each instance in the Repeated? This would solve the problem. Other suggestions are most welcome.

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list = {1,3,2,0,1,0,1,2,0,1,1,4,0};

This is an interesting problem. I'm not sure how to solve it using Condition rather than PatternTest.