
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
Define  $f(x)=-\frac{2x}{4x+3}$  and  $g(x)=\frac{x+2}{2x+1}$ . Moreover, let  $h^{n+1} (x)=g(f(h^n(x)))$ , where  $h^1(x)=g(f(x))$ . If the value of  $\sum_{k=1}^{100} (-1)^k\cdot h^{100}(k)$  can be written in the form  $ab^c$ , for some integers  $a,b,c$  where  $c$  is as maximal as possible and  $b\ne 1$ , find  $a+b+c$ .

*Proposed by **AOPS12142015***

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