
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
  
For any given positive integer \( m \), define an infinite sequence \(\{a_n\}\ (n \geq 1)\): \( a_1 \) is a positive integer. For any \( n \geq 1 \),  
\[ a_{n+1} = \begin{cases} 
a_n^2 + 2^m, & \text{if } a_n < 2^m \\
\frac{a_n}{2}, & \text{if } a_n \geq 2^m. 
\end{cases} \]  

For each \( m \), determine all possible values of \( a_1 \) such that all terms in \(\{a_n\}\) are integers.

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