
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 $a_1=a$ ($a>0$), $b_1-a_1=1$, $b_2-a_2=2$, $b_3-a_3=3$, if the sequence $\{a_n\}$ is unique, then $a= \boxed{\frac{1}{3}}$.

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