
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
For the sequence $\{a_n\}$, define $\Delta a_n$ as the first-order difference sequence of $\{a_n\}$, where $$\Delta a_{n}=a_{n+1}-a_{n} \quad (n\in\mathbb{N}^{*})$$, for a natural number $k$, define $\Delta^{k}a_n$ as the $k$-th order difference sequence of $\{a_n\}$, where $$\Delta^{k}a_{n}=\Delta^{k-1}a_{n+1}-\Delta^{k-1}a_{n}.$$  
(1) If $\Delta a_n=2$ and $a_1=1$, then $a_{2013}=$ \_\_\_\_\_\_ ;  
(2) If $a_1=1$ and $$\Delta^{2}a_{n}-\Delta a_{n+1}+a_{n}=-2^{n} \quad (n\in\mathbb{N}^{*})$$, then the general formula for the sequence $\{a_n\}$ is \_\_\_\_\_\_ .

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