
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 2019 Women's Volleyball World Cup, the Chinese women's volleyball team won the championship with an excellent record of 11 consecutive victories, presenting a great gift to the motherland on her 70th birthday. The volleyball matches adopt a 5-set 3-win system, with the first 4 sets using a 25-point system. Each team must win at least 25 points and lead the opponent by at least 2 points to win a set. In the deciding set (fifth set), a 15-point system is used, and each team must win at least 15 points and lead the opponent by 2 points to win. In each set, the serving team earns 1 point if they win the rally and retains the serve for the next rally; otherwise, the serve is exchanged, and the opponent earns 1 point. Now, two teams, Team A and Team B, are playing a volleyball match:
- $(1)$ If after the first three sets, Team A has won two sets and Team B has won one set. The probability of each team winning each subsequent set is $\frac{1}{2}$. Find the probability of Team A winning the entire match.
- $(2)$ If after the first four sets, Team A and Team B have each won two sets. In the deciding set (fifth set), both teams have 14 points each, and Team A has the next serve. The probability of Team A scoring 1 point when serving is $\frac{2}{5}$, and the probability of Team A scoring 1 point when receiving the serve is $\frac{3}{5}$. The team that scores retains the serve for the next rally. Let $x$ be the number of rallies played before Team A wins the match, where $x \leq 4$. Find the possible values of $x$ and the corresponding probability $P(x)$.

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