
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 of the function

\[ f(x)=\begin{cases} 
\operatorname{arctg}\left(x^{2} \cdot \sin \left(\frac{1}{5x}\right)\right), & x \neq 0, \\
0, & x = 0
\end{cases} \]

and calculate its value at the point \( x_0 = 0 \).

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