
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
The rectangular region \( A \) is in the first quadrant. The bottom side and top side of \( A \) are formed by the lines with equations \( y = 0.5 \) and \( y = 99.5 \), respectively. The left side and right side of \( A \) are formed by the lines with equations \( x = 0.5 \) and \( x = 99.5 \), respectively.

(a) Determine the number of lattice points that are on the line with equation \( y = 2x + 5 \) and are inside the region \( A \). (A point with coordinates \((r, s)\) is called a lattice point if \( r \) and \( s \) are both integers.)

(b) For some integer \( b \), the number of lattice points on the line with equation \( y = \frac{5}{3}x + b \) and inside the region \( A \) is at least 15. Determine the greatest possible value of \( b \).

(c) For some real numbers \( m \), there are no lattice points that lie on the line with equation \( y = mx + 1 \) and inside the region \( A \). Determine the greatest possible real number \( n \) that has the property that, for all real numbers \( m \) with \( \frac{2}{7} < m < n \), there are no lattice points on the line with equation \( y = mx + 1 \) and inside the region \( A \).

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