
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
During the Spring Festival, a mall decided to select 3 products for a promotion event from among 3 types of clothing, 2 types of home appliances, and 3 types of daily necessities.  
(1) Calculate the probability that at least one of the 3 selected products is a home appliance.  
(2) The mall promotes a selected product through a lottery method, where the price of the product is raised by 100 yuan, and customers have 3 chances to win a lottery: if they win once, they receive a prize of m yuan; if they win twice, the total prize is 3m yuan; if they win three times, the total prize is 6m yuan. Assuming the probability of winning each lottery is $$\frac {1}{3}$$, what is the maximum amount m such that the promotion is profitable for the mall?

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