
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
A positive integer is called fancy if it can be expressed in the form

$$
2^{a_{1}}+2^{a_{2}}+\cdots+2^{a_{100}},
$$

where $a_{1}, a_{2}, \ldots, a_{100}$ are non-negative integers that are not necessarily distinct.

Find the smallest positive integer $n$ such that no multiple of $n$ is a fancy number.

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