
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 function $f(x)=\frac{(\sin x-\cos x)\sin 2x}{\sin x}$.
(I) Find the domain and the minimum positive period $T$ of $f(x)$;
(II) Find the range of $x$ for which $f(x)\geqslant 0$;
(III) Determine if there exists a minimum positive real number $m$ such that the graph of $f(x)$ becomes an even function after shifting it left by $m$ units. If such $m$ exists, find its value; otherwise, explain the reason.

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