
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
How many sequences of nonnegative integers  $a_1,a_2,\ldots, a_n$  ( $n\ge1$ ) are there such that  $a_1\cdot a_n > 0$ ,  $a_1+a_2+\cdots + a_n = 10$ , and  $\prod_{i=1}^{n-1}(a_i+a_{i+1}) > 0$ ?

*Ray Li.*

<details><summary>Clarifications</summary>[list=1][*]If you find the wording of the problem confusing, you can use the following, equivalent wording: "How many finite sequences of nonnegative integers are there such that (i) the sum of the elements is 10; (ii) the first and last elements are both positive; and (iii) among every pair of adjacent integers in the sequence, at least one is positive."[/list]</details>

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