
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
For reals  $x\ge3$ , let  $f(x)$  denote the function
\[f(x) = \frac {-x + x\sqrt{4x-3} } { 2} .\]Let  $a_1, a_2, \ldots$ , be the sequence satisfying  $a_1 > 3$ ,  $a_{2013} = 2013$ , and for  $n=1,2,\ldots,2012$ ,  $a_{n+1} = f(a_n)$ . Determine the value of
\[a_1 + \sum_{i=1}^{2012} \frac{a_{i+1}^3} {a_i^2 + a_ia_{i+1} + a_{i+1}^2} .\]

*Ray Li.*

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