
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 an arithmetic sequence $\{a_n\}$ whose sum of the first $n$ terms is $S_n$, and a geometric sequence $\{b_n\}$ with the first term 1 and common ratio $q$, where $S_2=a_3=b_3$, and $a_1, a_3, b_4$ form a geometric sequence.  
(1) Find the general formulas for $\{a_n\}$ and $\{b_n\}$.  
(2) Let the sum of the first $n$ terms of $\{b_n\}$ be $T_n$. If $2S_n - na_n = b + \log_a (2T_n + 1)$ holds for all positive integers $n$, find the values of the real numbers $a$ and $b$.

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