
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  $a_1,a_2,\cdots,a_{41}\in\mathbb{R},$  such that  $a_{41}=a_1, \sum_{i=1}^{40}a_i=0,$  and for any  $i=1,2,\cdots,40, |a_i-a_{i+1}|\leq 1.$  Determine the greatest possible value of $(1)a_{10}+a_{20}+a_{30}+a_{40};$  $(2)a_{10}\cdot a_{20}+a_{30}\cdot a_{40}.$ 

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