
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 quadratic function $f(x) = x^2 - 2\sin\theta x + \frac{1}{4}$, (where $\theta \in \mathbb{R}$),
1. If $\theta = \frac{\pi}{6}$, find the range of the function $f(x)$ when $x \in [-1, 1]$.
2. If the function $f(x)$ is monotonic on the interval $\left[-\frac{1}{2}, \frac{1}{2}\right]$, determine the set of possible values for $\theta$.
3. If for any $x_1, x_2 \in [2, 3]$, it always holds that $|f(x_1) - f(x_2)| \leq 2\sin\theta t^2 + 8t + 5$ for any $\theta \in \mathbb{R}$, find the range of values for $t$.

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