
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
On the coordinate plane, a point whose $x$- and $y$-coordinates are both integers is said to be a 'lattice point'. If a lattice point is randomly chosen inside (excluding the boundary) the square with vertices $(0,0)$, $(101,0)$, $(101,101)$, and $(0,101)$, what is the probability that the line segment (excluding the two endpoints) joining $(0,2010)$ and the point chosen contains an even number of lattice points?

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