
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
A certain entertainment program has three levels of challenges. The rules are as follows: contestants participate in the first, second, and third levels successively, with the prize money for each level being $600, $900, and $1500 respectively. The prize money can be accumulated; if a contestant successfully completes a level, they can choose to end the game and receive the corresponding prize money, or they can choose to continue to the next level. If a contestant fails a level, all the prize money earned so far will be reset to zero, and the game ends. Contestant Xiao Li participates in this challenge. It is known that the probabilities of him successfully completing the first, second, and third levels are $\frac{3}{4}$, $\frac{2}{3}$, and $\frac{1}{2}$ respectively. The probability of Xiao Li choosing to continue after successfully completing the first level is $\frac{3}{5}$, and after successfully completing the second level is $\frac{2}{5}$. The success or failure of each level is independent of the others.  
$(1)$ Find the probability that Xiao Li successfully completes the first level but ends up with zero total prize money.  
$(2)$ Let $X$ be the random variable representing the total prize money Xiao Li receives. Find the probability distribution and the expected value of the random variable $X$.

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