Coin flipping puzzle - conditional expectations ("Pizza gameshow") [on hold]

Coin flipping puzzle - conditional expectations ("Pizza gameshow") [on hold]

This is my first post here, apologies if the question is trivial. I did
try searching for versions of this, I didn't see any relevant posts.
I found this puzzle online and haven't been able to convince myself of a
solution yet. Here goes:
A game show works as follows: The host calls N people at random. He flips
two fair coins, if they come up Heads Heads, the N people receive a pizza
and the show ends. For any other outcome, the host repeats the game, but
with 2*N different people. You sit at home one evening and receive a call
from the host who informs you that you've been chosen for the current
round. What is the chance of you receiving a pizza?
Some clarifications: There are enough people to call and pizzas to
distribute. Nobody can be called twice. You weren't watching and have no
idea what round you are in.
Is the answer 25% or 50%?
If it's 25%, what about the following reasoning: When the game ends after
round N, over 50% of the people who received a call will have received a
pizza. The value converges to 50% as N -> infinity. No matter when the
game ends, at least 50% of the population will receive a pizza.
If I know that half the people who receive a call receive a pizza, why
would I assume that I only have a 25% chance?
Lest anyone says that the problem lies in my unbounded population size or
number of turns: Suppose I limit the number of coin-tossing rounds to M,
and if we reach round M+1, X% of the people in this round receive a pizza.
Regardless of whether X is 0 or 100.0, once M is large enough, the
expectation value is close to 50% -- well below 25%. Is it still true that
if I receive a call (knowing the limit M) I only have a 25% chance of
receiving a pizza?
Does the answer change if I am told that I will receive a call before the
game starts? (i.e., is the relevant difference whether I know in advance
that I will be part of the game vs whether I find out as I pick up the
phone?) Does the answer change if the coin toss sequence has been done in
advance, and calling people only happens after that? Does knowing which
round I am in (but not the total number) matter?
Sorry for the verbose phrasing, but I'm having a hard time pinning down
where exactly the two answers diverge -- is there a subtlety in the
framing of the question that would make 50%/25% correct? Is this a case of
improper priors? Is the question ill-defined?
I realize that the expected number of pizzas delivered is infinite, but
that is not the case if we fix an upper bound for the number of rounds
played, yet the problem still remains.