
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  $f:\mathbb{N}\mapsto\mathbb{R}$  be the function \[f(n)=\sum_{k=1}^\infty\dfrac{1}{\operatorname{lcm}(k,n)^2}.\] It is well-known that  $f(1)=\tfrac{\pi^2}6$ .  What is the smallest positive integer  $m$  such that  $m\cdot f(10)$  is the square of a rational multiple of  $\pi$ ?

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