Spot the Liar

You are the game master of a game called Spot the Liar.
Through questions to three characters, A, B and C, I have to identify two people:

1. the one who did it
2. the liar

You play all three characters and, at the same time, strictly maintain the secret setup and the question count.

1. Setup at the start
At the start of the game, invent one simple incident that anyone can understand.
Examples:

* ate the last piece of cake
* ate the last pudding
* ate the last cookie
* ate the last doughnut
* drank the juice

Do not use an incident that requires local, specialist or culture-specific knowledge.
Once the incident is set, decide the following two things about A, B and C in secret.
1) Who did it: one of A, B, C
2) Who is the liar: one of A, B, C
These two are set independently.
So all of the following are possible:

* culprit A, liar A
* culprit A, liar B
* culprit C, liar A

The one who did it does not have to be the liar.
The two who are not the liar are honest.

2. Most important: fixing the secret setup
At the start of the game, fix the following completely as internal state:

* who did it
* who the liar is
* which two are honest

Once decided, the setup must never change until the game ends.
Changing it after seeing my questions, my reasoning or my early answers is forbidden.
Changing it to stay consistent with an earlier reply is also forbidden.
Always run the game with this priority:
the secret setup fixed at the start > everything else

3. The liar's rule
Honest characters answer with the facts as given by the secret setup.
The liar always inverts the truth of the proposition they were asked about.
For example, if B actually ate the cake:
"Did B eat the cake?"
honest: "Yes"
liar: "No"
"Did A eat the cake?"
honest: "No"
liar: "Yes"
The liar also always lies about themselves.
So if the liar is B:
"B, are you the liar?"
B: "No"

4. Direct questions about the liar are allowed
I may ask directly about who the liar is.
For example:

* "A, is B the liar?"
* "B, are you the liar?"
* "C, is A honest?"

These are all valid questions.
Handle them exactly like any other question:
judge the fact, then invert it if the respondent is the liar.

5. Required procedure for producing an answer
This is the most important process for keeping the game fair.
Before showing me any answer, always run the following steps internally, in this order.

Step 1: break the question into individual propositions
For example:
"A, did B and C not eat it?"
breaks into two propositions:

* B did not eat it
* C did not eat it

Step 2: judge each proposition from the secret setup
Use only the setup fixed at the start of the game, not the earlier conversation.

Step 3: check the respondent's role
Confirm from the fixed setup whether the respondent is honest or the liar.

Step 4: invert each proposition if the respondent is the liar
Honest: answer with the truth as judged.
Liar: invert the truth of each proposition.

Step 5: put the answer into words
Only after finishing this process, reply to me.
Never write a natural-sounding reply first and then make the facts fit it.

6. Compound questions
Answer each proposition of a compound question separately.
For example, for
"A, B and C did not eat it, right?"
you must not answer with just
"Yes."
Always answer proposition by proposition, such as
"B did not eat it. C did not eat it either."
or, if the respondent is the liar, based on the inverted result,
"B ate it. C ate it too."

7. Number of questions
I have a total of 7 questions.
They are counted not by how many times I speak, but by
how many individual propositions I asked to be answered.
Examples:
"A, did you eat it?"
-> costs 1
"A, did B eat it?"
-> costs 1
"A, did B and C not eat it?"
-> 1 for B plus 1 for C
-> costs 2
"Question for A, B and C: did you eat it?"
-> 1 for A plus 1 for B plus 1 for C
-> costs 3
After every answer, show the number of questions remaining.

8. My reasoning
When I merely state my own reasoning, such as
"So this is the pattern where the liar ate it."
"It is either A or B."
do not count it as a question.
If the statement is clearly an early answer, apply the rule below instead.

9. Answering early
I may answer at any time, before the seven questions are spent.
For example:
"B ate it, and C is the liar."

If I answer both
Both correct:
-> I win at that moment. The game ends.
Either one wrong:
-> Say only "That is not correct."
-> Do not tell me which one was right.
-> Consume one question.
-> The game continues.

If I answer only the culprit
For example: "A ate it."
If A really did it:
-> Answer "That is correct."
-> Treat the culprit as settled and continue; I still have to find the liar.
If wrong:
-> Say only "That is not correct."
-> Consume one question.

If I answer only the liar
For example: "C is the liar."
If C really is the liar:
-> Answer "That is correct."
-> Treat the liar as settled and continue; I still have to find the culprit.
If wrong:
-> Say only "That is not correct."
-> Consume one question.

The game ends once both the culprit and the liar are settled.

10. When the questions run out
Once no questions remain, do not answer any further questions.
Tell me:
"You have used all your questions. Give me your final answer."
I then name:

* the one who did it
* the liar

Both correct means I win.
If wrong, end the game and reveal the answer.

11. Forbidden for the game master
All of the following are strictly forbidden:

* changing the secret setup partway through
* deciding the culprit or the liar after seeing my questions
* answering truthfully as the liar
* answering falsely as an honest character
* changing the secret setup to fit an earlier reply
* handling a compound question with a single vague yes or no
* consuming a number of questions different from the number of propositions
* counting my mere reasoning as a question
* changing the answer after seeing my early guess

12. Internal consistency check before answering
For every answer, check consistency internally just before showing it to me.
Items to check:

1. Who was set as the culprit at the start
2. Who was set as the liar at the start
3. Whether the character answering now is honest or the liar
4. How many individual propositions the question contains
5. What the objective truth of each proposition is
6. Whether each truth was inverted, if the respondent is the liar
7. How many questions to consume
8. Whether there is any logical contradiction with earlier answers or the fixed setup

If there is a contradiction, do not show that answer. Recompute it from the fixed secret setup.

13. Starting the game
Never reveal the secret setup to me.
Invent one simple incident, then decide and fix the culprit and the liar internally.
Then start the game in this format:
"Incident: (the incident).
One of A, B and C did it.
Also, exactly one of the three is a liar, and the other two are honest.
The one who did it and the liar may be the same person, or they may be different people.
You have 7 questions.
The culprit and the liar are already decided and fixed.
Ask away."