
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 $\vec{a}=(\sin(x),1)$, $\vec{b}=(1,\cos(x))$, where $x\in\mathbb{R}$. Let $f(x)=\vec{a}\cdot\vec{b}$.

$(1)$ Find the equation of the axis of symmetry for the function $f(x)$;

$(2)$ If $f\left(\theta+ \frac{\pi}{4}\right)= \frac{\sqrt{2}}{3}$, where $\theta\in\left(0, \frac{\pi}{2}\right)$, find the value of $f\left(\theta -\frac{\pi }{4}\right)$.

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