
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
If the graph of the quadratic function $y=x^{2}+bx+c$ passes through points $A\left(0,2\right)$ and $B\left(1,0\right)$.<br/>$(1)$ Find the values of $b$ and $c$.<br/>$(2)$ Express the analytical expression of this quadratic function in the form of $y=a\left(x-h\right)^{2}+k$, where $h=$______, $k=______.

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