
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
(1) The sequence $\{a\_n\}$ satisfies the relation $a\_n a_{n+1} = 1 - a_{n+1}$ for all $n \in \mathbb{N^*}$, and $a_{2010} = 2$. Find $a_{2008}$.

(2) The sequence $\{a\_n\}$ is defined by $a\_1 = 1$ and $a_{n+1} = 2a\_n + 1$ for all $n \in \mathbb{N^*}$. Find the general term formula for $\{a\_n\}$.

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