I have a radioactive particle in a box, prepared so as to initially be in a pure state

$\psi\_0 =1\ \theta\_U+ 0\ \theta\_D$

(U is Undecayed, D is Decayed). 
I put a Geiger counter in the box.

Over time (t), the theory says that the state should evolve into a pure state that is a superposition of Undecayed and Decayed, with the Decayed part getting bigger and bigger

$\psi\_t =a\ \theta\_U+ b\ \theta\_D$

Eventually the counter will 'click', indicating that the particle has Decayed. Now I know that the state is 100% Decayed.

However, before this happened, the *silence* of the counter also indicated that the particle hadn't Decayed yet. So all the time up to that point, I also knew that the state was 100% Undecayed.

But this would be contradicting what the theory suggests (a superposition with a non zero contribution of the Decayed state, after some time), so I'm guessing it's an incorrect way of analysing the experiment.

I want to know where the mistake lies.

In other words, it seems to me the Geiger counter is *always* measuring the state of the particle. Silence means Undecayed, click means Decayed. So the particle would never actually Decay since I continuously know its state is 

$\psi\_t =1\ \theta\_U+ 0\ \theta\_D$

which means its chance of decaying would be perpetually zero (Zeno's effect, I've heard?).

How do I deal with this constant 'passive' measuring?