
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 $m$, $n\in N^*$, define $f_n(m) = \frac{n(n-1)(n-2)\ldots(n-m+1)}{m!}$.
(1) Let $a_m = f_6(m)$, find the value of $a_1 + a_2 + \ldots + a_{12}$;
(2) Let $b_m = (-1)^m m f_n(m)$, find the set of all possible values for $b_1 + b_2 + \ldots + b_{2n}$.

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