
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 lattice point in an \(xy\)-coordinate system defined as any point \((x, y)\) where both \(x\) and \(y\) are integers. The graph of the equation \(y = mx + 3\) passes through no lattice points with \(1 \leq x \leq 150\) for all \(m\) such that \( \frac{1}{3} < m < b\). What is the maximum possible value of \(b\)?
- **A)** \(\frac{51}{151}\)
- **B)** \(\frac{51}{152}\)
- **C)** \(\frac{50}{149}\)
- **D)** \(\frac{50}{150}\)
- **E)** \(\frac{52}{153}\)

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