
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 quadratic function $y=-x^{2}+bx+c$ passes through points $A(-1,0)$ and $B(3,0)$.  
$(1)$ Find the values of $b$ and $c$, as well as the coordinates of the vertex of this graph.  
$(2)$ If point $P(m,n)$ lies on the graph of this quadratic function and the distance to the $y$-axis is less than $2$, find the range of values for $n$.

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