
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 company has two research and development teams, Team A and Team B, with success probabilities of developing new products being $\frac{2}{3}$ and $\frac{3}{5}$, respectively. Team A is assigned to develop new product A, and Team B is assigned to develop new product B. It is assumed that the research and development of Teams A and B are independent of each other.  
$(1)$ Find the probability of at least one new product being successfully developed.  
$(2)$ If new product A is successfully developed, the company is expected to make a profit of $120$ thousand dollars; if new product B is successfully developed, the company is expected to make a profit of $100$ thousand dollars. Find the distribution table and mean profit of the company.

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