How do I find the equivalent capcitance of this infinite ladder?

Attempt:

Since $||- ||$ 1-2 system in series is repeating infinitely, I replaced it with $C_{eq}$

Then I got $||-||$ 1-2 system in parallel with $C_{eq}$

$\implies C_{eq} = \dfrac 23 + C{eq} \implies 0 = \dfrac 23$ which is obviously wrong.

Please let me know the flaw in my argument and please provide a hint on how to solve this question.

edited May 23 by sammy gerbil