
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
Facing a certain type of influenza virus, medical research institutions in various countries are researching vaccines. There are currently three independent research institutions, $A$, $B$, and $C$, with probabilities of developing vaccines within a certain period being $\frac{1}{5}$, $\frac{1}{4}$, and $\frac{1}{3}$, respectively. Find:<br/>
$(1)$ The probability that they all develop vaccines;<br/>
$(2)$ The probability that at least one of them develops a vaccine;<br/>
$(3)$ The probability that at most one institution develops a vaccine.

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