
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
Let \( T = \left\{ |9^k| \mid k \in \mathbf{Z}, 0 \leqslant k \leqslant 4000 \right\} \). It is known that \( 9^{4000} \) is a 3817-digit number, and its first digit is 9. How many numbers in \( T \) have 9 as their first digit?

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