
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
Two people, A and B, each shoot once at targets A and B. If a target is hit twice, it will be destroyed. The probability of A hitting targets A and B is both $\frac{1}{2}$, while the probability of B hitting targets A and B is $\frac{1}{3}$ and $\frac{2}{5}$, respectively.  
$(1)$ Find the probability of target A being destroyed.  
$(2)$ Find the probability of exactly one target being destroyed.

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