
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. The probability of success for developing a new product by Team A is $\frac{4}{5}$, and for Team B is $\frac{3}{4}$. Team A is assigned to develop a new product $A$, and Team B is assigned to develop a 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 that exactly one new product is successfully developed.  
$(2)$ If the development of new product $A$ is successful, the company will make a profit of $150$ thousand dollars, otherwise it will incur a loss of $60$ thousand dollars. If the development of new product $B$ is successful, the company will make a profit of $120$ thousand dollars, otherwise it will incur a loss of $40$ thousand dollars. Find the probability distribution and the mathematical expectation $E(\xi)$ of the company's profit (in thousand dollars).

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