
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
Consider a pentagonal array of numbers, where the edges start with $0, 1, 2, 3, \dots$ and each inner number is obtained by adding the three closest adjacent numbers on the previous row. The corners of the pentagon are considered as summation of two numbers for the inner pentagonal layer but does not change for the external layer. The first two non-trivial layers (inner layers) are:

\[
\begin{array}{ccccccccccccccc}
& & & & & & 0 & & & & & & & & \\
& & & & & 1 & & 1 & & 1 & & & & & \\
& & & & 2 & & 2 & & 2 & & 2 & & 2 & & \\
& & & 3 & & 5 & & 5 & & 5 & & 5 & & 3 & \\
& & 4 & & 9 & & 10 & & 10 & & 10 & & 9 & & 4 
\end{array}
\]

Find the sum of the numbers in the 100th layer, where numbers are generated according to the rules described.

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