
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
Consider a set of lattice points $T$ in the coordinate plane, where both coordinates of the points are integers between $1$ and $40$, inclusive. Exactly $400$ points in $T$ are located on or below a line with the equation $y = nx$. Calculate the length of the interval of all possible values of $n$ in the simplified form $\frac{p}{q}$, where $p$ and $q$ are coprime integers. What is the value of $p+q?$
A) 9
B) 11
C) 14
D) 17
E) 21

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