
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 the function $f(x) = \ln{x} + \cos{x} - \left(\frac{6}{\pi} - \frac{9}{2}\right)x$ with derivative $f'(x)$, and a sequence $\{a_n\}$ such that $a_{n+1} + a_{n} = nf'\left(\frac{\pi}{6}\right) + 3$ for $n \in \mathbb{N}^*$.

1. If the sequence $\{a_n\}$ is an arithmetic sequence, find the value of $a_1$.
2. When $a_1 = 2$, find the sum of the first $n$ terms of the sequence $S_n$.
3. If for any $n \in \mathbb{N}^*$, the inequality $\frac{a_n^2 + a_{n+1}^2}{a_n + a_{n+1}} \geq 4$ holds, find the range of possible values for $a_1$.

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