python rock paper scissors game

so i was just making a python rock paper scissors game and it has a score limit of 5. so if either the user or the computer reaches the score of 5, the game will print the person won or the computer won respectively. but it prints it if the computer wins only and also when the user wins the game doesn’t end and it doesn’t print that the user won. pls help me this is my code:

import random

user_score = int(0)
pc_score = int(0)
score_limit = 5


while user_score != score_limit or pc_score != score_limit:

    user_guess: str = input(str("pls enter your guess, rock/paper/scissors:")).lower()

    my_list = "rock", "paper", "scissors"
    pc_guess = random.choice(my_list)
    print("the computer chose", pc_guess)

    if pc_guess == "rock" and user_guess == "rock":
        print ( "Tie!!" )

    if pc_guess == "paper" and user_guess == "paper":
        print ( "Tie!!" )

    if pc_guess == "scissors" and user_guess == "scissors":
        print ( "Tie!!" )

    if pc_guess == "paper" and user_guess == "rock":
        print ( "the computer scores" )

        pc_score = pc_score + 1
        print ( "the pc score is:", pc_score )

    if pc_guess == "rock" and user_guess == "paper":
        print ( "yay u scored" )

        user_score = user_score + 1
        print ( "the user score is:", user_score )

    if pc_guess == "rock" and user_guess == "scissors":
        print ( "the computer scored" )

        pc_score = int ( pc_score ) + 1
        print ( "the pc score is:", pc_score )

    if pc_guess == "scissors" and user_guess == "rock":
        print ( "yay u scored" )

        user_score = user_score + 1
        print ( "the user score is:", user_score )

    if pc_guess == "paper" and user_guess == "scissors":
        print ( "yay u scored" )

        user_score = user_score + 1
        print ( "the user score is:", user_score )

    if pc_guess == "scissors" and user_guess == "paper":
        print ( "the computer scored" )

        pc_score = int ( pc_score ) + 1
        print ( "the pc score is:", pc_score )

        pc_guess = random.choice ( my_list )

        if user_score == score_limit:
            print("yay congrats u won")

        elif pc_score == score_limit:
            print("oopsies, the computer won, better luck next time")

            break