
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
Let  $ y_0$  be chosen randomly from  $ \{0, 50\}$ , let  $ y_1$  be chosen randomly from  $ \{40, 60, 80\}$ , let  $ y_2$  be chosen randomly from  $ \{10, 40, 70, 80\}$ , and let  $ y_3$  be chosen randomly from  $ \{10, 30, 40, 70, 90\}$ . (In each choice, the possible outcomes are equally likely to occur.) Let  $ P$  be the unique polynomial of degree less than or equal to  $ 3$  such that  $ P(0) \equal{} y_0$ ,  $ P(1) \equal{} y_1$ ,  $ P(2) \equal{} y_2$ , and  $ P(3) \equal{} y_3$ . What is the expected value of  $ P(4)$ ?

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