
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
Define the minimum real  $C$  where for any reals  $0 = a_0 < a_{1} < \dots < a_{1000}$  then   $$ \min_{0 \le k \le 1000} (a_{k}^2 + (1000-k)^2) \le C(a_1+ \dots + a_{1000}) $$   holds. Find  $\lfloor 100C \rfloor.$ 

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