
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 team composed of three people named A, B, and C won a prize of 40,000 yuan. They decided to distribute the prize money as follows: A and B would randomly take either 10,000 yuan or 20,000 yuan as their prize, each with a probability of P<sub>1</sub> for 10,000 yuan and a probability of P<sub>2</sub> for 20,000 yuan. The remaining prize money would go to C.
(1) If P<sub>1</sub>=P<sub>2</sub>=$$\frac{1}{2}$$, calculate the probability that C gets 10,000 yuan in prize money.
(2) If the expected prize money for A, B, and C is equal, find the values of P<sub>1</sub> and P<sub>2</sub>.

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