
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 a ping-pong match played to 11 points, each point won gives 1 point, and after a tie at 10:10, the service alternates for each point. The first player to lead by 2 points wins the game and ends the match. Players A and B are competing, and the probability of A scoring a point when serving is 0.5, while the probability of A scoring a point when B serves is 0.4. The results of each point are independent. After a tie at 10:10 in a game, A serves first, and they play X more points to finish the game.

(1) Calculate P(X=2);
(2) Calculate the probability of the event "X=4 and A wins".

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