
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 vectors $\overrightarrow{a}=(\sin x,\cos x)$ and $\overrightarrow{b}=(\sin x,\sin x)$, define the function $f(x)= \overrightarrow{a}\cdot \overrightarrow{b}$.
$$(1)$$ Find the equation of the axis of symmetry for $f(x)$;
$$(2)$$ Find the interval(s) where $f(x)$ is monotonic;
$$(3)$$ If for any real number $x\in\left[ \frac{\pi}{6}, \frac{\pi}{3}\right]$, the inequality $f(x)-m < 2$ always holds true, find the range of real numbers $m$.

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