Given a fixed enumeration of Infinite Time Turing Machines (ITTMs), let $M_i(x)$ denote a computation of an $i$-th ITTM, assuming that the input $x$ is a real (an infinite binary sequence).

Then the function $f(i)$ (input: natural number, output: countable ordinal) is defined as follows:

1. if $M_i(x)$ does not halt for any real $x$, then $f(i) = 0$;
2. if there exists the largest countable ordinal $\alpha$ such that there exists a real $x$ such that $M_i(x)$ halts at time $\alpha$ (and there does not exist a real $y \neq x$ such that $M_i(y)$ halts at time $\beta > \alpha$), then $f(i) = \alpha$;
3. if both (1) and (2) are false (that is, if for any countable ordinal $\alpha$ there exists a real $x$ such that $M_i(x)$ halts at time $\beta > \alpha$), then $f(i) = 0$.

The ordinal $\tau$ is defined as the supremum of the infinite set $\{f(0), f(1), f(2), \ldots \}$. Question: how large is $\tau$? In particular, what is $\tau$ in comparison with the least $\Sigma_1$-stable ordinal (mentioned in the Definition 3.1 in the paper “Recognizable sets and Woodin cardinals: Computation beyond the constructible universe”)?