
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
Find the derivative \( y_x'(x) \) of the function given by \( x(t) = \operatorname{ctg}(t^2) \) and \( y(t) = \operatorname{arcctg}(t^4) \).

\[ 
y_{x}^{\prime}(x) = \frac{\left(\operatorname{arcctg}(t^4)\right)_{t}^{\prime}}{\left(\operatorname{ctg}(t^2)\right)_{t}^{\prime}} = \frac{-\frac{1}{1+t^{8}} \cdot 4 t^{3}}{-\frac{1}{\sin^2(t^2)} \cdot 2 t} = \frac{2 t^{2} \cdot \sin^2(t^2)}{1+t^{8}}
\]

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