
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
Find threshold values $\mu_{100}$ and $\mu_{1 000000}$ such that 

$$
F = m n \sin \frac{\pi}{m} \sqrt{\frac{1}{n^{2}} + \sin^{4} \frac{\pi}{m}}
$$

is definitely greater than 100 and 1,000,000 respectively, for all $m$ greater than $\mu_{100}$ and $\mu_{1 000000}$, assuming $n = m^{3}$. (It is not necessary to provide the smallest such $\mu$ values.)

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