
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
Let $a$, $b$, $m$ be positive integers. If $a$ and $b$ have the same remainder when divided by $m$, then $a$ and $b$ are said to be congruent modulo $m$. Denote this as $a \equiv b \pmod{m}$. Given $a=2+2\times3+2\times3^2+\ldots+2\times3^{2003}$, and $b \equiv a \pmod{3}$, the value of $b$ can be \_\_\_\_\_\_ (write down all the numbers that satisfy the condition)  
①1007；②2013；③3003；④6002．

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