
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 $\Delta ABC$, $3\overrightarrow{BD}=2\overrightarrow{DC}$, $\overrightarrow{AD}\cdot \overrightarrow{BC}=0$, and the area of $\Delta ABC$, $S=\frac{3}{5}{{\left| BC \right|}^{2}}$. Determine the range of the function $y=2\sin x+\sqrt{2}\cos (x+\angle BAC)$, where $0\leqslant x\leqslant \frac{\pi }{2}$.

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