
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 $f(x) = [x \cdot [x]]$, where $[x]$ represents the largest integer not greater than $x$. When $x \in [0, n) (n \in \mathbb{N}^*)$, let the set of values of the function $f(x)$ be set $A$, and the number of elements in set $A$ be $a_n$. The minimum value of $\frac{a_n + 49}{n}$ is $\_\_\_\_\_\_\_\_\_\_.$

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