
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
\\((1)\\) The interval of increase for the function \\(y=\left(\frac{1}{2}\right)^{x^2-4x+3}\\) is \_\_\_\_\_\_

\\((2)\\) The graph of the function \\(y=\log_a(x+1)+2, (a>0, a\neq 1)\\) always passes through a fixed point, which is \_\_\_\_\_\_.

\\((3)\\) In triangle \\(ABC\\), with sides opposite to angles \\(A, B, C\\) being \\(a, b, c\\) respectively, if \\(\frac{a}{\cos A}=\frac{b}{\cos B}=\frac{c}{\cos C}\\), then triangle \\(ABC\\) is a \_\_\_\_\_\_ triangle.

\\((4)\\) Given a regular tetrahedron \\(ABCD\\) with all four vertices on the surface of a sphere centered at \\(O\\), and point \\(P\\) is the midpoint of edge \\(BC\\) with \\(BC=6\sqrt{2}\\), a plane passing through point \\(P\\) intersects the sphere \\(O\\). The minimum area of the intersection is \_\_\_\_\_\_

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