
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 of a line $y = mx + 5$ in an $xy$-coordinate system, where each coordinate $(x, y)$ has integer values. The line should not cross any lattice points for integer $x$ values within $10 < x \leq 200$. Determine the maximum value of $a$ for the slope $m$ such that $1 < m < a$ ensures the line avoids all these lattice points.
- **A)** $\frac{101}{100}$
- **B)** $\frac{102}{101}$
- **C)** $\frac{103}{100}$
- **D)** $\frac{103}{102}$

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