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To solve this problem, we can use Python code. The steps are as follows:

1. Ask the user for input of the specific number.
2. Ask the user for input of the numbers.
3. Use a loop to count the occurrences of each number after the specific number.
4. Use another loop to find the number with the highest count.
5. Print the result.

Here is the Python code:

num = int(input("Enter the specific number: "))
input_list = input("Enter the numbers separated by spaces: ").split()

count_dict = {}

for i in range(len(input_list) - 1):
    if int(input_list[i]) == num:
        if input_list[i+1] not in count_dict:
            count_dict[input_list[i+1]] = 1
        else:
            count_dict[input_list[i+1]] += 1

most_frequent = None
highest_count = 0
for key, value in count_dict.items():
    if value > highest_count:
        highest_count = value
        most_frequent = key

print("The number that occurs most frequently after", num, "is", most_frequent)

To use this code, simply copy and paste it into a Python IDE or text editor and run it.