
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 \(a_{1}+a_{2}+a_{3}+\ldots+a_{99}+a_{100}=0\). Find:

\(1 \cdot\left(a_{1}-a_{2}\right)+2\left(a_{2}-a_{3}\right)+3\left(a_{3}-a_{4}\right)+\ldots+99\left(a_{99}-a_{100}\right)+100 a_{100}\).

Extract the common factor and simplify the following expressions if possible:
1. \(a\left(a^{5}-a^{4}\right) a\)
2. \(2^{17}-2^{16}\)
3. \(a^{4}-\left|-a^{2}\right|\)
4. \((2 a-2)^{5}\)
5. \(6^{7}-3^{7}\)
6. \(a^{m+n}-a^{n}\)

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