
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 $\overrightarrow{a}=(\sqrt{3}\sin x,\cos x)$, $\overrightarrow{b}=(\cos x,\cos x)$  
$(1)$ If $\overrightarrow{a}\cdot \overrightarrow{b}=1$, and $x\in\left[-\frac{\pi}{4}, \frac{\pi}{4}\right]$, find the value of $x$;  
$(2)$ Let $f(x)= \overrightarrow{a}\cdot \overrightarrow{b}$, find the period and the interval of monotonic decrease for $f(x)$.

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