
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
Given a function $y=f(x)$, if there exists an interval $[m,n]$ that simultaneously satisfies the following conditions:
1) $f(x)$ is monotonic on $[m,n]$;
2) When the domain is $[m,n]$, the range of $f(x)$ is also $[m,n]$, then $[m,n]$ is called the "harmonious interval" of the function. Determine the range of values for the real number $a$ such that the function $f(x)=\frac{a+1}{a}-\frac{1}{x} (a > 0)$ has a "harmonious interval".

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