
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 that $\{a_n\}$ is a geometric sequence, and the sum of its first $n$ terms is $S_n$, it is known that $$S_n = 2^n + a$$ for all $n \in \mathbb{N}^*$.  
(1) Find the value of $a$ and the general term formula of the sequence $\{a_n\}$.  
(2) Let $b_n = \log_4 (a_n) + 1$ and denote the sum of the first $n$ terms of $\{b_n\}$ as $S'_n$. Find the solution set of the inequality $2S'_n \leq 5$.

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