
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 $y=-x^{2}+bx+c$ (where $b$ and $c$ are constants) passes through the points $(0,-3)$ and $(-6,-3)$.  
$(1)$ Find the values of $b$ and $c$.  
$(2)$ Find the maximum value of $y$ when $-4\leqslant x\leqslant 0$.

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