
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
A line is defined by the equation $y = mx + 3$ in an $xy$-coordinate system, where $x$ and $y$ are integers (lattice points). For what maximum value of $m$ does the graph pass through no lattice points with $0 < x \le 200$ if $\frac{1}{3} < m < a$?

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