
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 that the analytical expression of the function $f(x)$ satisfies $f(x+1)= \frac {x^{2}+2x+a+1}{x+1}\;(a > 0)$.

1. Find the analytical expression of the function $f(x)$;
2. When $a=1$, determine the monotonicity of the function $f(x)$ on the interval $(0,+∞)$, and provide a proof;
3. When $a=1$, let the function $g(x)= \begin{cases} f(x),x > 0 \\ f(-x)\;,x < 0 \end{cases}$, find the range of the function $g(x)$ on the interval $[-2,- \frac {1}{2}]$.

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