
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 equation $y = mx + 3$ on an $xy$-coordinate system, where each point $(x, y)$ has integer coordinates. Determine the maximum value of $a$ such that the graph of the line does not intersect any lattice points for any $m$ such that $\frac{1}{3} < m < a$ when $1 \leq x \leq 50$.
**A)** $\frac{1}{3}$  
**B)** $\frac{17}{51}$  
**C)** $\frac{1}{2}$  
**D)** $\frac{18}{53}$  
**E)** $\frac{16}{47}$

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