
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 \(25\), inclusive. Exactly \(350\) points in \(T\) lie on or below a line with equation \(y = mx\). Additionally, exactly \(150\) points in \(T\) lie on or above a line with equation \(y = 25 - mx\). 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) 13
B) 14
C) 15
D) 16
E) 17

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