
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 the sequence $\{a_n\}$, $a_1=1$, $a_{2n}=a_n$, $a_{4n-1}=0$, and $a_{4n+1}=1$ for $n\in \mathbb{N}^*$.  
(1) Find $a_4$ and $a_7$.  
(2) Determine whether there exists a positive integer $T$ such that for any $n \in \mathbb{N}^*$, we have $a_{n+T}=a_n$.

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