
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
Given the function $f(x) = \ln \left( \frac{ex}{e-x} \right)$, if $f\left( \frac{e}{2013} \right) + f\left( \frac{2e}{2013} \right) + \ldots + f\left( \frac{2012e}{2013} \right) = 503(a+b)$, then the minimum value of $a^{2}+b^{2}$ is \_\_\_\_\_\_.

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