
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
At the year-end summary meeting of a network technology company, in order to add joy and harmony to the atmosphere, a checkpoint game was designed, which required passing through several checkpoints to succeed. The first checkpoint is a question-answering stage, with three questions set: "General Knowledge of Literature and History," "General Knowledge of Life," and "General Knowledge of Film and Art." Two answer options are provided:<br/>Option 1: Answer all three questions, with at least two correct answers;<br/>Option 2: Randomly select two out of the three questions and answer them correctly.<br/>By completing either Option 1 or Option 2, the first checkpoint can be passed. Assuming that Programmer A and Programmer B have the probability $p\left(p\in \left(0,1\right)\right)$ of answering each of the three questions correctly, and the correctness of the three questions does not affect each other. Programmer A chooses Option 1, and Programmer B chooses Option 2.<br/>$(1)$ Calculate the probability of Programmer A and Programmer B passing the first checkpoint respectively;<br/>$(2)$ Let the number of people passing the first checkpoint among Programmer A and Programmer B be $\xi $, is there a unique value $p_{0}$ of $p$ such that $E\left(\xi \right)=1$? Explain the reason.

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