
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 two geometric sequences $\{a_n\}$ and $\{b_n\}$, satisfying $a_1=a$ ($a>0$), $b_1-a_1=1$, $b_2-a_2=2$, and $b_3-a_3=3$.  
(1) If $a=1$, find the general formula for the sequence $\{a_n\}$.  
(2) If the sequence $\{a_n\}$ is unique, find the value of $a$.

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