
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 $\overrightarrow {a}=(\sin x,\cos x)$, $\overrightarrow {b}=(\sin x,\sin x)$, and the function $f(x)=\overrightarrow {a}\cdot \overrightarrow {b}$.

(I) Find the equation of the axis of symmetry of $f(x)$.

(II) Determine the set of values of $x$ for which $f(x)\geqslant 1$.

(III) If the inequality $f(x)-m \lt 2$ holds for all real numbers $x\in [\dfrac{\pi }{6}, \dfrac{\pi }{3}]$, find the range of real numbers for $m$.

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