
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 that the graph of the quadratic function $y=x^{2}+bx+c$ passes through the points $(0,-1)$ and $(2,7)$.  
$(1)$ Find the analytical expression and axis of symmetry of the quadratic function.  
$(2)$ If the points $(-5,y_{1})$ and $(m,y_{2})$ are two different points on the parabola, and $y_{1}+y_{2}=28$, find the value of $m$.

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