
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
Consider a piece of paper turned upside down. The numbers 0, 1, and 8 remain the same when rotated $180^\circ$, 6 becomes 9, and 9 becomes 6. Other numbers are meaningless when rotated $180^\circ$. 

How many 7-digit numbers remain unchanged when rotated $180^\circ$? Out of these, how many are divisible by 4? What is the sum of all these 7-digit numbers that remain unchanged when rotated $180^\circ$?

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