
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
Let  $P_0P_5Q_5Q_0$  be a rectangular chocolate bar, one half dark chocolate and one half white chocolate, as shown in the diagram below. We randomly select  $4$  points on the segment  $P_0P_5$ , and immediately after selecting those points, we label those  $4$  selected points  $P_1, P_2, P_3, P_4$  from left to right. Similarly, we randomly select  $4$  points on the segment  $Q_0Q_5$ , and immediately after selecting those points, we label those  $4$  points  $Q_1, Q_2, Q_3, Q_4$  from left to right. The segments  $P_1Q_1, P_2Q_2, P_3Q_3, P_4Q_4$  divide the rectangular chocolate bar into  $5$  smaller trapezoidal pieces of chocolate. The probability that exactly  $3$  pieces of chocolate contain both dark and white chocolate can be written as  $\frac{m}{n}$ , where  $m$  and  $n$  are relatively prime positive integers. Find  $m + n$ .
[Diagram in the individuals file for this exam on the Chmmc website]

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