
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
Purchasing a certain type of insurance, each policyholder pays a premium of $a$ yuan per year to the insurance company. If the policyholder has a claim during the year of purchase, they can receive a claim payment of 10,000 yuan. Assuming that 10,000 people bought this insurance during the year, and that the risks for each policyholder are independent of each other. It is known that the probability of the insurance company having to pay at least 10,000 yuan in claim payments during the year is $1 - 0.999^{104}$.
(Ⅰ) Find the probability $p$ of a policyholder having a claim within a year;
(Ⅱ) Assuming that the insurance company's cost of operating this type of insurance, excluding claim payments, is 50,000 yuan, to ensure the expected profit is not less than zero, find the minimum premium each policyholder should pay (unit: yuan).

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