
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
A function $f(x)$ is defined for all real numbers $x$. For all non-zero values of $x$, the relationship is given by
\[ 2f\left(x\right) + f\left(\frac{1}{x}\right) = 3x + 6 \]
Let $T$ denote the sum of all the values of $x$ for which $f(x) = 3010$. Compute the integer nearest to $T$.

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