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REPL-like output

Hello, Sage Community.

If I run the sage REPL in a terminal, and execute something like

for i in range(4):
    print(i^2)

then we all know that Sage prints

In this case, I could create a class to intercept standard output to a variable, like:

class verboseBuffer:
    def __init__(self):
        self.buffer = ''
    def write(self, s):
        self.buffer += s

Then I just make

_verbOut_ = verboseBuffer()
sys.stdout = _verbOut_

and if I run the first chunk of code, then the integers will be stored in _verbOut_.buffer, instead of being printed to standard output as before. I finally can do

sys.stdout = sys.__stdout__
print(_verbOut_.buffer)

and this prints the contents of _verbOut_.buffer, i.e., the list at the beginning.

Now consider the same loop without the print:

for i in range(4):
    i^2

Executed in the REPL, Sage will print exactly the same as before (even if I haven't specified to print), and I could do the same thing to redirect the standard output to a variable.

Now, on the other hand, I have a huge file for which I would like to print everything in the same way it is printed in the REPL. I can't add print to every single part of the file, so I have tried to redirect output to a variable as before, and print it in the last line. The problem is that, things like the last chunk of code, without an explicit print won't be stored in _verbOut_.buffer at the end of the execution.

My questions are:

Why does the Sage REPL prints all computations not assigned to a variable, like the i^2 in the loop, but that doesn't happen if the code is in a file?
How can I obtain a REPL-like output without having to add print to everything in my long file?