
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
After learning the application of the multiplication formula "$\left(a\pm b\right)^{2}=a^{2}\pm 2ab+b^{2}$", Teacher Wang raised a question: Find the minimum value of the algebraic expression $x^{2}+4x+5$. Students were required to use their knowledge to solve the problem. After exploring, exchanging, and discussing, the students summarized the following solution method:

Solution: $x^{2}+4x+5=x^{2}+4x+2^{2}-2^{2}+5=\left(x+2\right)^{2}+1$,
$\because \left(x+2\right)^{2}\geqslant 0$, $\therefore \left(x+2\right)^{2}+1\geqslant 1$.
When $\left(x+2\right)^{2}=0$, the value of $\left(x+2\right)^{2}+1$ is the smallest, and the minimum value is $1$.
Therefore, the minimum value of $x^{2}+4x+5$ is $1$.
Please answer the following questions based on the above method:
$(1)$ Write down the minimum value of $\left(x-1\right)^{2}+3$ as ______.
$(2)$ Find the minimum value of the algebraic expression $x^{2}+10x+32$.
$(3)$ Do you think the algebraic expression $-\frac{1}{3}{x^2}+2x+5$ has a maximum value or a minimum value? Find the maximum or minimum value.
$(4)$ If $7x-x^{2}+y-11=0$, find the minimum value of $x+y$.

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