
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 \( f(n) \) denote the number of 0's in the decimal representation of the positive integer \( n \). For example, \( f(10001123) = 3 \) and \( f(1234567) = 0 \). Define the expression:
\[
M = f(1) \times 2^{f(1)} + f(2) \times 2^{f(2)} + f(3) \times 2^{f(3)} + \cdots + f(99999) \times 2^{f(99999)}
\]
Find the value of \( M - 100000 \).

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