
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 that $|\vec{a}| = |\vec{b}| = 1$ and $|\vec{a} + k\vec{b}| = \sqrt{3}|k\vec{a} - \vec{b}|$ ($k > 0$), let $f(k) = \vec{a} \cdot \vec{b}$.  
$(1)$ Find $f(k) = \vec{a} \cdot \vec{b}$ in terms of $k$;  
$(2)$ When $k > 0$, $f(k) \geqslant x^{2} - 2tx - \frac{5}{2}$ holds true for any $t \in [-2,2]$, find the range of the real number $x$.

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