
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 all lattice points in the coordinate plane where both coordinates are integers from 1 to 40, inclusive. Exactly 600 points in \( T \) lie on or below a line with equation \( y = mx \). The possible values of \( m \) lie in an interval of length \( \frac{a}{b} \), where \( a \) and \( b \) are relatively prime positive integers. What is \( a+b \)?
**A)** 7
**B)** 8
**C)** 9
**D)** 10
**E)** 11

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