
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 $A_n^5 = 56C_n^7$, and $(1-2x)^n = a_0 + a_1x + a_2x^2 + a_3x^3 + \ldots + a_nx^n$.

$(I)$ Find the value of $n$;

$(II)$ Find the value of $a_1 + a_2 + a_3 + \ldots + a_n$;

$(III)$ Find the value of $|a_1| + |a_2| + |a_3| + \ldots + |a_n|$.

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