
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
In April 2022, the new coronavirus epidemic touched the hearts of the people across the country. According to the requirements of the higher authorities, a certain city needs to select 4 experts in nursing surgery and psychological therapy from a certain People's Hospital in the city to join the provincial expert group to rush to Shanghai for rescue work. The hospital currently has 3 nursing experts $A_{1}$, $A_{2}$, $A_{3}$, 5 surgical experts $B_{1}$, $B_{2}$, $B_{3}$, $B_{4}$, $B_{5}$, and 2 psychological therapy experts $C_{1}$, $C_{2}$.

$(1)$ Find the number of ways to select 4 individuals with 1 surgical expert and 1 psychological therapist.

$(2)$ Find the number of ways to select at least 2 surgical experts, where $B_{1}$ and $A_{1}$ cannot be selected simultaneously.

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