
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
In the Cartesian coordinate system, the graph of the quadratic function $y=ax^{2}+bx+c\left(a \lt 0\right)$ passes through the points $\left(m,b\right)$ and $\left(m+1,a\right)$. When $b\geqslant a$ and $m \lt 0$, the graph of the quadratic function $y=ax^{2}+bx+c$ has a maximum value of $-2$. The maximum value of $a$ is ______.

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