
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 sequence $\{a_n\}$ where $a_1=1$, and the point $(a_n, a_{n+1})$ lies on the graph of the function $y=x+1$ $(n\in\mathbb{N}^*)$, and the sequence $\{b_n\}$ is a geometric sequence with all positive terms, and $b_2=2, b_4=8$.

(1) Find the general formula for the sequences $\{a_n\}$ and $\{b_n\}$;

(2) If the sequence $\{c_n\}$ satisfies $c_n=(-1)^n a_n + b_n$, and let $T_n$ be the sum of the first $n$ terms of the sequence $\{c_n\}$, find the value of $T_{100}$.

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