
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 about $x$, $y=a\left(x-m\right)^{2}+a\left(x-m\right)$, where $a$ and $m$ are constants with $a \gt 0$ and $m \gt 0$. The graph of the function intersects the $x$-axis at points $A$ and $B$. Please complete the following questions:<br/>$(1)$ The length of segment $AB$ is ______.<br/>$(2)$ If the vertex of the graph of this quadratic function is $C$, and it intersects the positive $y$-axis at point $D$. When ${S_{\triangle ABC}}=\frac{1}{3}{S_{\triangle ABD}}$, the value of $m$ is ______.

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