
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  $a_1,a_2,\ldots,a_6$  be a sequence of integers such that for all  $1 \le i \le 5$ ,  $$ a_{i+1}=\frac{a_i}{3} \quad \text{or} \quad a_{i+1}={-}2a_i. $$  Find the number of possible positive values of  $a_1+a_2+\cdots+a_6$  less than  $1000$ .

*Proposed by **stayhomedomath***

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