
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
If $(1+mx)^{8}=a_{0}+a_{1}x+a_{2}x^{2}+\cdots+a_{8}x^{8}$, where $a_{3}=56$. Find:<br/>
$(1)$ the value of $m$;<br/>
$(2)$ $a_{1}+a_{2}+\cdots +a_{8}$;<br/>
$(3)$ $({a}_{0}+{a}_{2}+{a}_{4}+{a}_{6}+{a}_{8})^{2}-({a}_{1}+{a}_{3}+{a}_{5}+{a}_{7})^{2}$.

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