
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
Find the maximum possible value of \( a \) such that the line \( y = mx + 3 \) passes through no lattice point with \( 0 < x \leq 150 \) for all \( m \) such that \( \frac{1}{3} < m < a \).
**A)** \( \frac{51}{152} \)  
**B)** \( \frac{52}{153} \)  
**C)** \( \frac{51}{151} \)  
**D)** \( \frac{50}{149} \)  
**E)** \( \frac{50}{150} \)

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