
When tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.

[ASSESS]

[ADVANCE]

[VERIFY]

[SIMPLIFY]

[SYNTHESIZE]

[PIVOT]

[OUTPUT]

You should strictly follow the format below:

[ACTION NAME]

# Your action step 1

# Your action step 2

# Your action step 3

...

Next action: [NEXT ACTION NAME]


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A certain Xinhua bookstore will hold a prize promotion activity on Children's Day. Children who purchase books at the bookstore up to a specified amount can participate in a two-person PK game to win "book vouchers." The rules of the game are as follows: There are three rounds in the game. Before each round, there are 5 balls with numbers 1, 2, 3, 4, 5 in an opaque box (the balls are identical except for the numbers). In each round, players A and B each draw a ball from the box without replacement (the player cannot see the number on the ball). If the difference between the two numbers drawn by the players is odd, player A loses 2 points, and player B gains 2 points. If the difference is even, player A gains n (n is a positive integer) points, and player B loses n points. At the beginning of the PK game, both player A and player B start with 0 points. After the PK game, if the points of both players are not equal, the player with more points wins and receives the "book voucher" reward. If the points are equal, neither player receives the reward.

$(1)$ Let the random variable $\xi$ represent player A's points after the PK game. Find the distribution of $\xi$.

$(2)$ Using the mathematical expectation of the random variable $\xi$ from part (1) as the basis for decision-making, when the game rules are more favorable for player A to receive the "book voucher" reward, let the minimum value of the positive integer n be denoted as $n_{0}$.

① Determine the value of $n_{0}$ and explain the reasoning.

② When $n = n_{0}$, calculate the probability that, with player A losing points in at least one round, player A still receives the "book voucher" reward.

Present the answer in LaTex format: \boxed{Your answer}
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