
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
Team A and Team B are playing in a basketball final, following a best-of-seven series where the first team to win four games wins the championship. Based on the previous match results, Team A's home and away game schedule is "home, home, away, away, home, away, home". It is given that the probability of Team A winning at home is $0.6$, and the probability of winning away is $0.5$. The results of each game are independent of each other. What is the probability of Team A winning with a score of $4:1$?

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