
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
Find the sum of the digits of the following product (as a function of \( n \)):
$$
9 \cdot 99 \cdot 9999 \cdot \cdots \cdot \left(10^{2^{n}} - 1\right).
$$

In this product, the number of digits of each factor is twice the number of digits of the previous factor.
(21st United States of America Mathematical Olympiad, 1992)

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