
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
Calculate $\Delta f\left(P_{0}\right)$ and $d f\left(P_{0}\right)$ for the function $f(x, y) = x^{2} y$ at the point $P_{0} = (5, 4)$ with $\Delta x = 0.1$ and $\Delta y = -0.2$.

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