
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 team match of a certain World Table Tennis Championship, the Chinese team will compete against the South Korean team. The match follows a best-of-5 format, where the first team to win 3 games wins. Based on past performance, the probability of the Chinese team winning each game is $\frac{3}{5}$.  
$(Ⅰ)$ Find the probability of the Chinese team winning with a score of $3:0$.  
$(Ⅱ)$ Find the probability of the Chinese team winning after losing the first game.  
$(Ⅲ)$ Assuming the total number of games in the match is a random variable $X$, find the probability distribution and the expected value $E(X)$ when the South Korean team wins the first game.

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