
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 school has two restaurants, $A$ and $B$. Wang randomly chooses one of the restaurants for lunch on the first day. If he goes to restaurant $A$ on the first day, the probability of him going to restaurant $A$ on the second day is $0.4$. If he goes to restaurant $B$ on the first day, the probability of him going to restaurant $A$ on the second day is $0.8$.  
$(1)$ Calculate the probability of Wang going to restaurant $A$ for lunch on the second day.  
$(2)$ Wang dines at restaurant $A$ once, which offers $4$ types of Western desserts and $6$ types of Chinese desserts. Wang chooses three types of desserts from these options. Let the number of Western desserts chosen be denoted as $X$. Find $P(X\geq 1)$.

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