unexpected object?

[urufu]

name = input("Your name, plz:")
file = "temp.txt"
with open(file, "w") as file:
    file.write(name + "\n")

file = "temp.txt"
while True:
    name = input("Your name, again:")
    if name == "":
        break
    else:
        with open(file, "a") as file:
            file.write(name + "\n")

When I input 3 names and it went wrong:
---------------------------------------------------------------------------
TypeError Traceback (most recent call last)
<ipython-input-5-44303ce02481> in <module>
10 break
11 else:
---> 12 with open(file, "a") as file:
13 file.write(name + "\n")

TypeError: expected str, bytes or os.PathLike object, not _io.TextIOWrapper
---------------------------------------------------------------------------

But I really don't know where the syntax error is...
Does anyone help me solve this issue? Thank you very much.