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Answer the question on the basis of the information given below:

Mathematicians are assigned a number called Zohos number (named after the famous mathematician, Paul Zohos). Only Paul Zohos himself has an Zohos number of zero. Any mathematician who has written a research paper with Zohos has an Zohos number of $1$. For other mathematicians, the calculation of his/her Zohos number is illustrated below:

Suppose that a mathematician $\text{X}$ has co-authored papers with several other mathematicians. From among them, mathematician $\text{Y}$ has the smallest Zohos number. Let the Zohos number of $\text{Y}$ be $y$. Then $\text{X}$ has an Zohos number of $y+1$. Hence any mathematician with no co-authorship chain connected to Zohos has an Zohos number of infinity.

In a seven day long mini-conference organized in memory of Paul Zohos, a close group of eight mathematicians, call them $\text{A, B, C, D, E, F, G and H}$, discussed some research problems. At the beginning of conference, $\text{A}$ was the only participant who had an infinite Zohos number. Nobody had an Zohos number less than that of $\text{F}$.

1. On the third day of the conference $\text{F}$ co-authored a paper jointly with $\text{A}$ and $\text{C}$. This reduced the average Zohos number of the group of eight mathematicians to $3$. The Zohos numbers of $\text{B, D, E, G and H}$ remained unchanged with the writing of this paper. Further, no other co-authorship among any three members would have reduced the average Zohos number of the group of eight to as low as $3$.
2. At the end of the third day, five members of this group had identical Zohos numbers while the other three had Zohos numbers distinct from each other.
3. On the fifth day, $\text{E}$ co-authored a paper with $\text{F}$ which reduced the group’s average Zohos number by $0.5$. The Zohos numbers of the remaining six were unchanged with the writing of this paper.
4. No other paper was written during the conference.

The Zohos number of $\text{C}$ at the end of the conference was: ________