
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 functions $f(x)=2ax^{2}-2bx-a+b$ and $g(x)=2ax-2b$, where $a,b \in \mathbb{R}$ and $a > 0$:

1. Find the maximum value of $f(\sin \theta)$ when $\theta \in [0, \frac{\pi}{2}]$.
2. If $|f(\sin \theta)| \leqslant 1$ holds for any $\theta \in \mathbb{R}$ and the maximum value of $g(\sin \theta)$ is $2$, find the expression for $f(x)$.

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