So none of us understood from your words. Different problem.
Picture below. Bottom trace is not Q1 but Q3 or 4 or ....
Look at Q1. It will change state on every falling edge. But we do not know which falling edge will cause Q2 to flip over.
Look at Original Clock and Q1. They will be in phase or our of phase. There will be some delay but they will be 99.5% together or 99.5% wrong.
So add a XOR gate looking at the two. The output of the XOR will be "good" for 99.5% or "bad" for 99.5% of the time. Add a RC on the output with a time constant to hide the small spikes (0.5% of the time) After the filter (RC) the output should be 1 or 0 all the time depending on which condition.
What I am thinking is to run the XOR (filtered) signal to the 4040's reset.
...On one mode the (XOR filtered) will all wise be high and the 4040 will count for ever.
...On the other mode the (XOR filtered) will be low and reset the 4040 so it might try again.
I know there is some thing wrong with my logic but think about it.
OR do what Alec_t said and forget this 2X thing. Your Original Clock = Q1 so just use Original Clock for Q1.