
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
Two circles $\mathcal{C}_{1}$ and $\mathcal{C}_{2}$ intersect at two distinct points $A$ and $B$. Let $I$ be the midpoint of the segment $[A B]$.

a) Is it possible to draw distinct lines concurrent at $I$ such that these lines and the circles $\mathcal{C}_{1}$ and $\mathcal{C}_{2}$ form exactly 2017 points of intersection? What about 2018?

b) What if $I$ is instead located anywhere within the area bounded by the intersection of $\mathcal{C}_{1}$ and $\mathcal{C}_{2}$ (but not on $[A B]$)?

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