
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 sequence ${a_n}$, let $S_n$ denote the sum of its first $n$ terms. The terms of the sequence ${a_n}$ are arranged according to the following rule: $\frac{1}{2}$, $\frac{1}{3}$, $\frac{2}{3}$, $\frac{1}{4}$, $\frac{2}{4}$, $\frac{3}{4}$, $\frac{1}{5}$, $\frac{2}{5}$, $\frac{3}{5}$, $\frac{4}{5}$... $\frac{1}{n}$, $\frac{2}{n}$,... $\frac{n-1}{n}$... If there exists a positive integer $k$ such that $S_{k-1} < 10$ and $S_{k} > 10$, find $a_{k}$ = \_\_\_\_\_\_.

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