
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 parabola $y=ax^{2}+bx+4$ where $a$ and $b$ are constants, $a\geqslant -1$, and $a\neq 0$, passing through point $A\left(1,1\right)$ and intersecting the $y$-axis at point $B$.<br/>$(1)$ Express $b$ in terms of $a$.<br/>$(2)$ When $a=-\frac{1}{2}$, find the axis of symmetry of the parabola.<br/>$(3)$ If the portion between points $A$ and $B$ on the parabola decreases from left to right, then the range of values for $a$ is ______.<br/>$(4)$ When $-1\leqslant x\leqslant 2$, the maximum value of the function $y=ax^{2}+bx+4$ (where $a\geqslant -1$ and $a\neq 0$) is $m$. Find the minimum value of $m$.

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