
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
Determine the value of a positive integer $a$ given that three positive integers $2a$, $1$, and $a^2+3$ can be arranged in an arithmetic sequence in some order.

(1) Find the value of $a$;
(2) If the arithmetic sequence $\{a_n\}$ has a first term and common difference both equal to $a$, and the geometric sequence $\{b_n\}$ has a first term and common ratio both equal to $a$, with the sums of the first $n$ terms denoted as $S_n$ for the arithmetic sequence and $T_n$ for the geometric sequence, and it is given that $\frac{T_n + 2}{2^n} > S_n - 130$, find the maximum value of positive integer $n$ that satisfies this condition.

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