
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
Define the sequence  $a_1,a_2,a_3,\ldots$  by  $a_n=\sum_{k=1}^n\sin(k)$ , where  $k$  represents radian measure. Find the index of the  $100$ th term for which  $a_n<0$ .

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