
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  $a_1, a_2, ..., a_{1999}$  be nonnegative real numbers satisfying the following conditions:
a.  $a_1+a_2+...+a_{1999}=2$ b.  $a_1a_2+a_2a_3+...+a_{1999}a_1=1$ .
Let  $S=a_1^ 2+a_2 ^ 2+...+a_{1999}^2$ . Find the maximum and minimum values of  $S$ .

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