
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 $\vec{m}=(1,a)$, $\vec{n}=(\sin x,\cos x)$ ($x\in \mathbb{R}$), and the function $f(x)=\vec{m}\cdot\vec{n}$, $\frac{\pi }{4}$ is a zero of the function $f(x)$.

  (1) Find the value of $a$ and the intervals where the function $f(x)$ is monotonically increasing;

  (2) If $\alpha ,\beta \in \left( 0,\frac{\pi }{2} \right)$, and $f\left( \alpha +\frac{\pi }{4} \right)=\frac{\sqrt{10}}{5}$, $f\left( \beta +\frac{3\pi }{4} \right)=\frac{3\sqrt{5}}{5}$, find the value of $\sin \left( \alpha +\beta \right)$.

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