
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
Let $[x]$ denote the greatest integer less than or equal to the real number $x$. Given:
$$
a_{k}=\sum_{i=k^{2}}^{(k+1)^{2}-1} \frac{1}{i} \quad (k=1,2, \cdots)
$$
Find:
$$
\sum_{k=1}^{n}\left(\left[\frac{1}{a_{k}}\right]+\left[\frac{1}{a_{k}}+\frac{1}{2}\right]\right) .
$$

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