
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
Let  $\mathbb{R}^*$  be the set of all real numbers, except  $1$ . Find all functions  $f:\mathbb{R}^* \rightarrow \mathbb{R}$  that satisfy the functional equation   $$ x+f(x)+2f\left(\frac{x+2009}{x-1}\right)=2010 $$  .

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