
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 a quadratic function in terms of $x$, $y_{1}=x^{2}+bx+c$ (where $b$ and $c$ are real numbers).<br/>$(1)$ If the graph of the quadratic function passes through the point $(0,4)$ and has a symmetry axis of $x=1$, find the expression of this quadratic function;<br/>$(2)$ If $b^{2}-c=0$ and the minimum value of the quadratic function is $21$ when $b-3\leqslant x\leqslant b$, find the value of $b$;<br/>$(3)$ Let the quadratic function in terms of $x$ be $y_{2}=2x^{2}+x+m$. Under the conditions in $(1)$, if $y_{2}\geqslant y_{1}$ for $0\leqslant x\leqslant 1$, find the minimum value of the real number $m$.

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