
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  $\dots, a_{-1}, a_0, a_1, a_2, \dots$  be a sequence of positive integers satisfying the folloring relations:  $a_n = 0$  for  $n < 0$ ,  $a_0 = 1$ , and for  $n \ge 1$ ,
\[a_n = a_{n - 1} + 2(n - 1)a_{n - 2} + 9(n - 1)(n - 2)a_{n - 3} + 8(n - 1)(n - 2)(n - 3)a_{n - 4}.\]
Compute
\[\sum_{n \ge 0} \frac{10^n a_n}{n!}.\]

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