
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 line given by $y = mx + b$ in an $xy$-coordinate system, which is composed of lattice points (points where both $x$ and $y$ are integers). Suppose $b$ is a known integer and a constant $T > 0$. The graph of $y = mx + b$ passes through no lattice point where $0 < x \leq T$ for all slopes $m$ such that $\frac{1}{3} < m < a$. Here, $m$ is written as $\frac{p}{q}$ in lowest terms where $q$ must not be a prime number. What is the maximum possible value of $a$ if $T = 150$ and $b=3$?
A) $\frac{152}{451}$
B) $\frac{153}{455}$
C) $\frac{154}{461}$
D) $\frac{155}{463}$

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