
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]


system
In the top 12 of the Asian qualifiers for the 2022 Qatar World Cup, the Chinese men's football team was eliminated in a disastrous manner with a record of 1 win, 3 draws, 6 losses, 9 goals scored, and 19 goals conceded. Football enthusiasts, A and B, decided to enhance their skills by practicing set-piece kicks. They take turns practicing (each person kicks once in a round) under the same conditions. In each round, A and B are at the same position, with one kicking the ball and the other attempting to save it. A kicks first, and each person kicks once. If one scores and the other does not, the scorer gets 1 point, and the non-scorer gets -1 point. If both score or both miss, they both get 0 points. It is given that the probability of A scoring each time is 1/2, the probability of B scoring each time is 2/3, the probability of A saving B's shot is 1/2, and the probability of B saving A's shot is 1/5. The outcomes of each kick are independent.

$(1)$ After one round of kicking, let $X$ denote A's score. Find the probability distribution and the expected value of $X$.

$(2)$ If after two rounds of kicking, $p_{2}$ represents the probability that A's cumulative score is higher than B's cumulative score. Find $p_{2}$.

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