def identify_pronouns_and_referents(sentence):
    """
    Identifies the pronouns and possible referents in the sentence.
    
    Example:
    Input: "Alice, Bob, and Claire are playing a game. At the start of the game, they are each holding a ball: Alice has a yellow ball, Bob has a blue ball, and Claire has a pink ball."
    Output: ({"they": ["Alice", "Bob", "Claire"]})
    """
    return pronouns_referents

def generate_sentences_with_referents(sentence, pronouns_referents):
    """
    Generates sentences by replacing the pronouns with each of the possible referents.
    
    Example:
    Input: ("Alice, Bob, and Claire are playing a game. At the start of the game, they are each holding a ball: Alice has a yellow ball, Bob has a blue ball, and Claire has a pink ball.", {"they": ["Alice", "Bob", "Claire"]})
    Output: ["Alice, Bob, and Claire are playing a game. At the start of the game, Alice is holding a ball: Alice has a yellow ball, Bob has a blue ball, and Claire has a pink ball.", ...]
    """
    return sentences

def check_sentences(sentences):
    """
    Checks if the generated sentences are logically correct and make sense.
    
    Example:
    Input: ["Alice, Bob, and Claire are playing a game. At the start of the game, Alice is holding a ball: Alice has a yellow ball, Bob has a blue ball, and Claire has a pink ball.", ...]
    Output: [["Alice", "yellow", "Bob", "blue", "Claire", "pink"]]
    """
    pass

def determine_referents(sentence, pronouns_referents):
    """
    Determines the referents for each pronoun or states that it's ambiguous.
    
    Example:
    Input: "Alice, Bob, and Claire are playing a game. At the start of the game, they are each holding a ball: Alice has a yellow ball, Bob has a blue ball, and Claire has a pink ball."
    Output: ["Alice", "yellow", "Bob", "blue", "Claire", "pink"]
    """
    generated_sentences = generate_sentences_with_referents(sentence, pronouns_referents)
    valid_sentences = check_sentences(generated_sentences)
    
    if len(valid_sentences) == 1:
        return valid_sentences[0]
    else:
        return "ambiguous"

def solve_game_problem(input_text):
    """
    Main function that integrates all the functions to solve the game problem.
    """
    parts = input_text.split("\n")
    sentence = parts[1].split("Sentence: ")[-1]
    options = parts[-3:]
    pronouns_referents = identify_pronouns_and_referents(sentence)
    referents = determine_referents(sentence, pronouns_referents)
    
    for option in options:
        if referents[-1] in option:
            return option.split()[0]