
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
Sequence of real numbers  $a_0,a_1,\dots,a_{1389}$  are called concave if for each  $0<i<1389$ ,  $a_i\geq\frac{a_{i-1}+a_{i+1}}2$ . Find the largest  $c$  such that for every concave sequence of non-negative real numbers:
\[\sum_{i=0}^{1389}ia_i^2\geq c\sum_{i=0}^{1389}a_i^2\]

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