
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 designed corresponding activity plans for two different activities, Activity A and Activity B. In order to understand whether the students at the school support the activity plans, a simple random sample of students was taken, and the data obtained is shown in the table below:

|           | Boys     |         | Girls    |         |
|-----------|----------|---------|----------|---------|
|           | Support  | Not Support | Support  | Not Support |
| Activity A| 200 people | 400 people | 300 people | 100 people |
| Activity B| 350 people | 250 people | 150 people | 250 people |

Assume that the students' support for the activity plans is independent of each other.

$(Ⅰ)$ Estimate the probability that boys at the school support Activity A and the probability that girls at the school support Activity A.

$(Ⅱ)$ Two boys are randomly selected from all the boys at the school, and one girl is randomly selected from all the girls at the school. Estimate the probability that exactly 2 out of these 3 students support Activity A.

$(Ⅲ)$ Let the estimated probability that students at the school support Activity B be denoted as $p_{0}$. Assume there are 500 boys and 300 girls in the first grade at the school. The estimated probability that students from other grades support Activity B is denoted as $p_{1}$. Compare the sizes of $p_{0}$ and $p_{1}$ without requiring a proof.

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