
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 $f(x) = |3x + \frac{1}{a}| + |x - a|$. 

(1) If $a = 1$, find the solution set for $f(x) \geqslant 8$.

(2) For any $a \in (0, \infty)$ and any $x \in \mathbb{R}$, if $f(x) \geqslant m$ always holds true, find the maximum value of the real number $m$.

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