
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 work group consisting of three people, A, B, and C, received a total of 40,000 yuan as a bonus. They decided to distribute the bonus in the following way: A and B will randomly take 10,000 yuan or 20,000 yuan as their own bonus, with the probability of taking 10,000 yuan being $P_1$ and the probability of taking 20,000 yuan being $P_2$. The remaining bonus will go to C.  
$(1)$ If $P_1=P_2= \frac{1}{2}$, find the probability that C gets 10,000 yuan as a bonus;  
$(2)$ If the expected values of the bonuses for A, B, and C are equal, find the values of $P_1$ and $P_2$.

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