
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
If the product of two natural numbers, when divided by 9, leaves a remainder of 1, we say that these two natural numbers are "modular inverses modulo 9." For example, \(2 \times 5 = 10\), and when divided by 9, it leaves a remainder of 1, so 2 and 5 are "modular inverses modulo 9." Similarly, \(1 \times 1 = 1\), so 1 is its own "modular inverse modulo 9." Clearly, if a natural number has a "modular inverse modulo 9," the inverse is not unique. For instance, 10 is also another "modular inverse modulo 9" of 1. Determine if the numbers 1, 2, 3, 4, 5, 6, 7, and 8 have a "modular inverse modulo 9" and list each number with its corresponding smallest "modular inverse modulo 9" if it exists.

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