
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 $Ox$ axis, the points $0, 1, 2, \ldots, 100$ are marked, and graphs of 200 different quadratic functions are drawn, each passing through two of the marked points and touching the line $y = -1$. For each pair of graphs, Oleg wrote on the board the number corresponding to how many points the graphs have in common. Then, he summed all 19900 numbers written on the board. Could he have obtained the number 39699?

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