
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 high school has 1000 students in the first year. Their choices of elective subjects are shown in the table below:

| Subject | Physics | Chemistry | Biology | Politics | History | Geography |
|---------|---------|-----------|---------|----------|---------|-----------|
| Number of Students | 300 | 200 | 100 | 200 | 100 | 100 |

From these 1000 students, one student is randomly selected. Let:
- $A=$ "The student chose Physics"
- $B=$ "The student chose Chemistry"
- $C=$ "The student chose Biology"
- $D=$ "The student chose Politics"
- $E=$ "The student chose History"
- $F=$ "The student chose Geography"

$(Ⅰ)$ Find $P(B)$ and $P(DEF)$.

$(Ⅱ)$ Find $P(C \cup E)$ and $P(B \cup F)$.

$(Ⅲ)$ Are events $A$ and $D$ independent? Please explain your reasoning.

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