
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
16. Create a cube  $C_1$  with edge length  $1$ . Take the centers of the faces and connect them to form an octahedron  $O_1$ . Take the centers of the octahedron’s faces and connect them to form a new cube  $C_2$ . Continue this process infinitely. Find the sum of all the surface areas of the cubes and octahedrons.

17. Let  $p(x) = x^2 - x + 1$ . Let  $\alpha$  be a root of  $p(p(p(p(x)))$ . Find the value of  $$ (p(\alpha) - 1)p(\alpha)p(p(\alpha))p(p(p(\alpha)) $$  18. An  $8$  by  $8$  grid of numbers obeys the following pattern:

1) The first row and first column consist of all  $1$ s.
2) The entry in the  $i$ th row and  $j$ th column equals the sum of the numbers in the  $(i - 1)$  by  $(j - 1)$  sub-grid with row less than i and column less than  $j$ .

What is the number in the 8th row and 8th column?

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