
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
Given the line equation \( y = mx + 3 \) plotted in the coordinate system, determine the maximum value of \( b \) such that the line does not intersect any lattice points for \( 0 < x \leq 200 \) if \( \frac{1}{3} < m < b \).

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