
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
The "Guess the Song Title" game is based on the ringtone made from the main melody of a song to guess the song title. The rules are as follows: Contestants guess the song title in order for the first, second, and third levels. If they successfully pass a level, they will respectively receive a charity fund of $1000, $2000, and $3000. After passing a level, a contestant can choose to end the game and take the corresponding charity fund, or continue to the next level. If they fail to pass any level, the game ends and all charity funds are reset to zero. Suppose a guest has probabilities of $\frac{3}{4}$, $\frac{2}{3}$, and $\frac{1}{2}$ for successfully passing the first, second, and third levels, respectively. The probability that the guest chooses to continue to the next level is $\frac{1}{2}$ for each level, and the success or failure of passing each level does not affect the others.  
$(1)$ Find the probability that the guest successfully passes the first level but receives zero charity fund.  
$(2)$ Find the probability distribution and mean of the total amount $X$ of charity funds the guest receives.

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