
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
Given that \(10^{-221} < M < 10^{-220}\) and \(10^{-2} < n < 10^{-1}\), and the mantissa of the common logarithm of \(M\) is \(n\), find the mantissa of the common logarithm of \(\frac{1}{M^{9}}\).

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