
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 $z=f(x, y)=x^{2}+y^{2}+2x+1$ and two points $A\left(x_{0}, y_{0}\right)=A(2, 3)$ and $B\left(x_{1}, y_{1}\right)=B(2.02, 2.99)$, it is required to:

1) Compute the value $z_{0}$ of the function $f(x, y)$ at point $A$ and the value $z_{1}$ at point $B$.
2) Compute the approximate value $\tilde{z}_{1}$ of the function at point $B$ based on the value $z_{0}$ at point $A$ by replacing the function's increment with the differential, and estimate the relative error percentage arising from this replacement.
3) Formulate the equation of the tangent plane to the surface $z=f(x, y)$ at point $C\left(x_{0}, y_{0}, z_{0}\right)$.

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