
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
Define the function $f(x) = 2ax + b$ on $\mathbb{R}$, where $a$ is a real number, and $b \in (0, +\infty)$. If for any $x \in [-\frac{1}{2}, \frac{1}{2}]$, the inequality $|f(x)| \leqslant 2$ always holds, then when $a \cdot b$ is at its maximum, the value of $f(2017)$ is _______ .

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