
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 a natural number \( a \geqslant 2 \), let \( A = \left\{ y \mid y = a^x, x \in \mathbf{N} \right\} \). On the interval \([1, a]\), does there exist \( b \) such that \( C = A \cap B \neq \emptyset \)? Here, \( B = \{ y \mid y = (a+1)x + b, x \in \mathbf{N} \} \). If such \( b \) exists, try to find all possible values of \( b \) and the corresponding set \( C \); if not, please explain why.

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