
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
Determine the maximum value of $a$ such that the line $y = mx + 3$ does not pass through any lattice points for $0 < x \leq 50$ and $\frac{1}{2} < m < a$.
**A)** $\frac{51}{101}$
**B)** $\frac{50}{99}$
**C)** $\frac{26}{51}$
**D)** $\frac{27}{53}$
**E)** $\frac{13}{25}$

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