
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
The terms of the arithmetic sequence $\{a_n\}$ are all positive, with $a_1=3$, and the sum of the first $n$ terms is $S_n$. The sequence $\{b_n\}$ is a geometric sequence, with $b_1=2$, and it is given that $b_2S_2=32$, $b_3S_3=120$.  
(1) Find $a_n$ and $b_n$;  
(2) Find the sum of the first $n$ terms, $T_n$, of the sequence $\{a_nb_n\}$;  
(3) If $\frac{1}{S_1} + \frac{1}{S_2} + \ldots + \frac{1}{S_n} \leq x^2 + ax + 1$ holds for any positive integer $n$ and any $x \in \mathbb{R}$, find the range of the real number $a$.

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