
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 $T$ be the set of lattice points in the coordinate plane, both of whose coordinates are integers between $1$ and $40$, inclusive. Exactly $800$ points in $T$ lie on or below a line with equation $y = nx$. The possible values of $n$ lie in an interval of length $\frac{c}{d}$, where $c$ and $d$ are relatively prime positive integers. What is $c+d?$
A) \( 17 \)
B) \( 19 \)
C) \( 21 \)
D) \( 23 \)
E) \( 25 \)

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