
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 largest real number  $\lambda$  such that

\[a_1^2 + \cdots + a_{2019}^2 \ge a_1a_2 + a_2a_3 + \cdots + a_{1008}a_{1009} + \lambda a_{1009}a_{1010} + \lambda a_{1010}a_{1011} + a_{1011}a_{1012} + \cdots + a_{2018}a_{2019}\]
for all real numbers  $a_1, \ldots, a_{2019}$ .  The coefficients on the right-hand side are  $1$  for all terms except  $a_{1009}a_{1010}$  and  $a_{1010}a_{1011}$ , which have coefficient  $\lambda$ .  

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