
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 brand specialty store is preparing to hold a promotional event during New Year's Day. Based on market research, the store decides to select 4 different models of products from 2 different models of washing machines, 2 different models of televisions, and 3 different models of air conditioners (different models of different products are different). The store plans to implement a prize sales promotion for the selected products, which involves increasing the price by $150$ yuan on top of the current price. Additionally, if a customer purchases any model of the product, they are allowed 3 chances to participate in a lottery. If they win, they will receive a prize of $m\left(m \gt 0\right)$ yuan each time. It is assumed that the probability of winning a prize each time a customer participates in the lottery is $\frac{1}{2}$.  
$(1)$ Find the probability that among the 4 selected different models of products, there is at least one model of washing machine, television, and air conditioner.  
$(2)$ Let $X$ be the random variable representing the total amount of prize money obtained by a customer in 3 lottery draws. Write down the probability distribution of $X$ and calculate the mean of $X$.  
$(3)$ If the store wants to profit from this promotional plan, what is the maximum amount that the prize money should be less than in order for the plan to be profitable?

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