
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 function $f(x)=x\cdot\lceil x \rceil$, where $\lceil x \rceil$ represents the smallest integer not less than $x$, such as $\lceil 1.5 \rceil = 2$, $\lceil -2.5 \rceil = -2$. When $x \in (0,n]$ and $n \in \mathbb{N}^*$, the range of the function $f(x)$ is denoted as $A_n$, and the number of elements in set $A_n$ is denoted as $a_n$. Calculate the sum $\frac{1}{a_1}+ \frac{1}{a_2}+ \ldots + \frac{1}{a_{10}}$.

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