
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
It is known that the sum of the first $n$ terms of an arithmetic sequence $\{a_n\}$ is $S_n$, $a_9 = \frac{1}{2}a_{12} + 6$, and $a_2 = 4$. Find the sum of the first $10$ terms of the sequence $\left\{\frac{1}{S_n}\right\}$.

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