
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 the graph of the equation \( y = mx + 3 \) in an \(xy\)-coordinate system. Determine the maximum possible value of \( b \) such that no lattice points are passed through by the line for \(0 < x \le 200\) when the slope \( m \) satisfies \(\frac{1}{3} < m < b\).
A) \(\frac{67}{201}\) 
B) \(\frac{68}{202}\) 
C) \(\frac{68}{203}\) 
D) \(\frac{69}{204}\)

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