
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
Let \\(\triangle ABC\\) have internal angles \\(A\\), \\(B\\), and \\(C\\) opposite to sides of lengths \\(a\\), \\(b\\), and \\(c\\) respectively, and it is given that \\(2\sin A\sin \left(C+ \dfrac{\pi}{6}\right)=\sin (A+C)+\sin C \).

\\((1)\\) Find the magnitude of angle \\(A\\).

\\((2)\\) If \\(b=2\\) and \\(c=1\\), and \\(D\\) is the midpoint of \\(BC\\), find the length of \\(AD\\).

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