
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
$$
f(x)=\frac{1+2x-x^{2}}{(1+x)\left(1+x^{2}\right)}
$$

Let \(\alpha, \beta, \gamma\) be the angles of an arbitrary acute triangle. Calculate the expression
$$
\begin{array}{l}
f(\tan \alpha)+f(\tan \beta)+f(\tan \gamma)+ \\
f(\cot \alpha)+f(\cot \beta)+f(\cot \gamma)=
\end{array}
$$

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