
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
For the equation \\(C:x^{2}+y^{2}-2x-4y+m=0\\) in terms of \\(x\\) and \\(y\\),

\\((1)\\) If equation \\(C\\) represents a circle, find the range of the real number \\(m\\);

\\((2)\\) When equation \\(C\\) represents a circle, if this circle intersects with the line \\(l:x+2y-4=0\\) at points \\(M\\) and \\(N\\), and \\(|MN|=\\dfrac{4\\sqrt{5}}{5}\\), find the value of the real number \\(m\\).

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