
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 $\Delta ABC$, let the sides opposite to angles $A$, $B$, and $C$ be $a$, $b$, and $c$ respectively, and $\sin \left(A-\frac{\pi }{6}\right)-\cos \left(A+\frac{5\pi }{3}\right)=\frac{\sqrt{2}}{2}$.

$(1)$ Find the magnitude of angle $A$;

$(2)$ If $a=\sqrt{5}$ and $\sin^2 B+\cos 2C=1$, find $b$ and $c$.

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