
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 the sequences $\{a_{n}\}$ and $\{b_{n}\}$, where $\{a_{n}\}$ is an arithmetic sequence with the sum of the first $n$ terms denoted as $S_{n}$, and let $\{b_{n}\}$ be a sequence with the sum of the first $n$ terms denoted as $T_{n}$, where $a_{5}=11$ and $S_{10}=120$, and $b_{n}=\frac{1}{{{a_n}•{a_{n+1}}}$, determine the value of the positive integer $k$ such that $T_{k}=\frac{1}{7}$.

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