
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
Find the smallest positive interger number  $n$  such that there exists  $n$  real numbers  $a_1,a_2,\ldots,a_n$  satisfied three conditions as follow:
a.  $a_1+a_2+\cdots+a_n>0$ ;
b.  $a_1^3+a_2^3+\cdots+a_n^3<0$ ;
c.  $a_1^5+a_2^5+\cdots+a_n^5>0$ .

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