
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 \\(f(x)=a(x-5)^2+6\ln x\\), where \\(a\in\mathbb{R}\\), the equation of the tangent line to the curve \\(y=f(x)\\) at the point \\((1,f(1))\\) is \\(2x-y+6=0\\).

\\((1)\\) Determine the value of \\(a\\);

\\((2)\\) Find the intervals of monotonicity and the extremum of the function \\(f(x)\\).

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