
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 $f\left(x\right)=\sin ^{2}\left(2x-\dfrac{\pi }{4}\right)-2t\cdot \sin \left(2x-\dfrac{\pi }{4}\right)+t^{2}-6t+1$ for $x\in [\dfrac{\pi }{24}, \dfrac{\pi }{2}]$, and its minimum value is $g\left(t\right)$.
1. If $t=1$, find the value of $f\left(\dfrac{\pi }{8}\right)$.
2. Find the expression for $g\left(t\right)$.
3. When $-\dfrac{1}{2}\leqslant t\leqslant 1$, in order for the equation $g\left(t\right)=kt$ to have a real root, find the range of real values for $k$.

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