
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 \( x \), we have
\[ 3f(x) + f\left(\frac{1}{x}\right) = 6x^2 + 5x + 3 \]
Let \( T \) denote the sum of all of the values of \( x \) for which \( f(x) = 3003 \). Compute the integer nearest to \( T \).

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