
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
In the Cartesian coordinate system $xOy$, it is known that the vertex of angle $\alpha$ is at the origin, the initial side coincides with the non-negative $x$-axis, and the terminal side passes through a point $P(-3,t)$.

(1) If $t=4$, find the value of $\sin\left(\alpha +\dfrac{\pi}{4}\right)$;

(2) If $t=\sqrt{3}$, and $\alpha \in (0,2\pi)$, find the interval of monotonicity of $f(x)=\sin(x+\alpha)+\cos x$.

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