
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
What is the smallest \( n \) for which there is a solution to the following system of equations:
\[
\sin x_1 + \sin x_2 + \cdots + \sin x_n = 0,
\]
\[
\sin x_1 + 2 \sin x_2 + \cdots + n \sin x_n = 100?
\]

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