
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 a fixed natural number \( n \):
1. Find all solutions to the equation 
   \[
   \sum_{k=1}^{n}\left\lfloor \frac{x}{2^{k}} \right\rfloor = x - 1
   \]
2. Given a natural number \( m \), determine the number of solutions to the equation 
   \[
   \sum_{k=1}^{n}\left\lfloor \frac{x}{2^{k}} \right\rfloor = x - m
   \]

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