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The problem states that we have subsets \( A_1, A_2, A_3 \) of the set \( \{1, 2, 3, 4, 5\} \), and they need to satisfy the condition:

\[
(A_1 \cup A_2) \cap (A_2 \cup A_3) = \{1, 2, 3, 4, 5\}.
\]

This means that every element in the set \( \{1, 2, 3, 4, 5\} \) must belong to both \( A_1 \cup A_2 \) and \( A_2 \cup A_3 \).

To satisfy this condition, each element in \( \{1, 2, 3, 4, 5\} \) must appear in either:
- \( A_1 \) or \( A_2 \), and
- \( A_2 \) or \( A_3 \).

### Analysis for Each Element
For any element \( x \in \{1, 2, 3, 4, 5\} \):
1. It can belong to both \( A_1 \) and \( A_3 \) independently.
2. It must be in at least one of \( A_1 \) or \( A_2 \) **and** at least one of \( A_2 \) or \( A_3 \).

This results in the following possible placements for each element \( x \):
1. \( x \in A_1 \) and \( x \in A_3 \).
2. \( x \in A_1 \) and \( x \notin A_3 \).
3. \( x \notin A_1 \) and \( x \in A_3 \).
4. \( x \in A_2 \) only (if \( x \) is not in either \( A_1 \) or \( A_3 \)).

Each element has 4 possible configurations to satisfy the condition.

### Total Configurations
Since there are 5 elements in \( \{1, 2, 3, 4, 5\} \), and each element can be assigned to \( A_1 \), \( A_2 \), and \( A_3 \) in 4 ways to satisfy the condition, the total number of ways to assign elements is:

\[
N = 4^5 = 1024.
\]

### Finding the Last Two Digits of \( N \)
To find the last two digits of \( 1024 \), we calculate \( 1024 \mod 100 \):

\[
1024 \mod 100 = 24.
\]

### Answer
The last two digits of \( N \) are \( \boxed{24} \).

Would you like more details on any part of the solution, or do you have further questions?

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Here are some related questions that expand on this concept:

1. What if the set had more elements, say \( \{1, 2, \dots, 6\} \)? How would the answer change?
2. How would the solution be affected if we required every element to appear in \( A_1 \) or \( A_3 \) but not necessarily in \( A_2 \)?
3. What if we changed the condition to \( (A_1 \cap A_2) \cup (A_2 \cap A_3) = \{1, 2, 3, 4, 5\} \)?
4. Can you generalize this approach to subsets \( A_1, A_2, \ldots, A_n \)?
5. How would the answer differ if we were only asked for the number of ways to create each subset without regard to the specific ordering of \( A_1, A_2, \) and \( A_3 \)?

**Tip:** When analyzing sets with union and intersection conditions, consider each element independently and account for the constraints it must satisfy across the sets.

Let A_1, A_2, A_3 be subsets of the set {1, 2, 3, 4, 5} such that (A_1 ∪ A_2) ∩ (A_2 ∪ A_3) = {1, 2, 3, 4, 5}. Let N be the number of ways to construct the list of sets (A_1, A_2, A_3). Find the last two digits of N.

Explore how to calculate the number of ways to construct subsets A1, A2, A3 of the set {1, 2, 3, 4, 5} that satisfy specific union and intersection conditions. This problem involves set theory and combinatorics concepts, suitable for advanced high school students in Grades 11-12.

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