
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
The Hangzhou Asian Games are underway, and table tennis, known as China's "national sport," is receiving a lot of attention. In table tennis matches, each game is played to 11 points, with one point awarded for each winning shot. In a game, one side serves two balls first, followed by the other side serving two balls, and the service alternates every two balls. The winner of a game is the first side to reach 11 points with a lead of at least 2 points. If the score is tied at 10-10, the service order remains the same, but the service alternates after each point until one side wins by a margin of 2 points. In a singles table tennis match between players A and B, assuming player A serves first, the probability of player A scoring when serving is $\frac{2}{3}$, and the probability of player A scoring when player B serves is $\frac{1}{2}$. The outcomes of each ball are independent. 

$(1)$ Find the probability that player A scores 3 points after the first 4 balls in a game.

$(2)$ If the game is tied at 10-10, and the match ends after X additional balls are played, find the probability of the event "X ≤ 4."

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