
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
Eve randomly chooses two  $\textbf{distinct}$  points on the coordinate plane from the set of all  $11^2$  lattice points  $(x, y)$  with  $0 \le x \le 10$ ,  $0 \le y \le 10$ . Then, Anne the ant walks from the point  $(0,0)$  to the point  $(10, 10)$  using a sequence of one-unit right steps and one-unit up steps. Let  $P$  be the number of paths Anne could take that pass through both of the points that Eve chose. The expected value of  $P$  is  $\dbinom{20}{10} \cdot \dfrac{a}{b}$  for relatively prime positive integers  $a$  and  $b$ . Compute  $100a+b$ .

*Proposed by Michael Tang*

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