
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 vectors $\overrightarrow{a}$ and $\overrightarrow{b}$ that satisfy $|\overrightarrow{a}| = |\overrightarrow{b}| = 1$ and $|k\overrightarrow{a} + \overrightarrow{b}| = \sqrt{3}|\overrightarrow{a} - k\overrightarrow{b}| (k > 0)$, let $f(k) = \overrightarrow{a} \cdot \overrightarrow{b}$.

1. Find $f(k) = \overrightarrow{a} \cdot \overrightarrow{b}$ (represent in terms of $k$);
2. If $f(k) \geq x^2 - 2tx - \frac{1}{2}$ holds true for any $k > 0$ and any $t \in [-1, 1]$, find the range of values for the real number $x$.

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