
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
Two players, A and B, are playing a table tennis match. It is known that the probability of A winning a game is $0.6$, and the probability of B winning a game is $0.4$. They agree to play a "best-of-3" match, where the first player to win $2$ games wins the match.  
$(1)$ Find the probability of A winning the match.  
$(2)$ Find the expected number of games won by A in this match (rounded to two decimal places).  
$(3)$ Based on the conclusion in $(2)$, calculate the variance of the number of games won by A in this match.

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