Four rectangular strips each measuring $4$ by $16$ inches are laid out with two vertical strips crossing two horizontal strips forming a single polygon which looks like a tic-tack-toe pattern. What is the perimeter of this polygon?

[asy]
size(100);
draw((1,0)--(2,0)--(2,1)--(3,1)--(3,0)--(4,0)--(4,1)--(5,1)--(5,2)--(4,2)--(4,3)--(5,3)--(5,4)--(4,4)--(4,5)--(3,5)--(3,4)--(2,4)--(2,5)--(1,5)--(1,4)--(0,4)--(0,3)--(1,3)--(1,2)--(0,2)--(0,1)--(1,1)--(1,0));
draw((2,2)--(2,3)--(3,3)--(3,2)--cycle);
[/asy]
1. To solve this problem, we need to understand the layout of the strips and how they form the polygon. Each strip is $4$ inches wide and $16$ inches long. When two vertical strips cross two horizontal strips, they form a tic-tac-toe pattern, creating a single polygon.

2. Let's visualize the layout:
   - The two vertical strips are placed parallel to each other, $4$ inches apart.
   - The two horizontal strips are also placed parallel to each other, $4$ inches apart.
   - The intersection of these strips forms a grid with $3$ vertical and $3$ horizontal segments.

3. To find the perimeter, we need to count the total length of the outer edges of the polygon. The outer edges consist of:
   - Two vertical segments of $16$ inches each on the left and right sides.
   - Two horizontal segments of $16$ inches each on the top and bottom sides.
   - Four shorter segments of $4$ inches each at the corners where the strips intersect.

4. Let's calculate the total perimeter step-by-step:
   - The two vertical segments contribute $2 \times 16 = 32$ inches.
   - The two horizontal segments contribute $2 \times 16 = 32$ inches.
   - The four shorter segments contribute $4 \times 4 = 16$ inches.

5. Adding these contributions together, we get:
   \[
   32 + 32 + 16 = 80 \text{ inches}
   \]

6. Therefore, the perimeter of the polygon is $80$ inches.

The final answer is $\boxed{80}$ inches.