
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 $f(x)=2x^{2}-4x+a$ and $g(x)=\log _{a}x$ ($a > 0$ and $a\neq 1$).

(I) Find the range of values for the real number $m$ such that the function $f(x)$ does not have monotonicity on $[-1,2m]$.

(II) If $f(1)=g(1)$.

  (i) Find the value of the real number $a$.

 (ii) Let $t_{1}= \frac {1}{2}f(x)$, $t_{2}=g(x)$, $t_{3}=2^{x}$. Compare the sizes of $t_{1}$, $t_{2}$, and $t_{3}$ when $x∈(0,1)$.

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