How can we find the value of beta for a transistor to enter in a saturation region?
If I would answer your question strictly, I would not care about input signal at all.
So you are not looking for working point (what would be practical), but just for value of beta which would take transistor in your circuit into saturation.

When transistor is in saturation Vbe is around 0.7V and Vce around 0.2V. (just ballpark values, these values differ from transistor to transistor).
Then:
\$Ib=(Vcc-Vbe)/Rb=47\mu A\$
\$Ic=(Vcc-Vce)/Rc=5.36 mA\$
\$\beta=Ic/Ib=113.9\$  

so the transistor has to have value of beta at least this value