
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 fixed point $M(-1,2)$, a line $l_{1}: y = a(x+1)$, and a curve $C: y = \sqrt{x^2 + 1}$, the line $l_{1}$ intersects $C$ at points $A$ and $B$. Let $N$ be the midpoint of the segment $AB$. The line $l_{2}$ passes through points $M$ and $N$, and its intercept on the $x$-axis is $m$. Express $m$ as a function of $a$: $m = f(a)$. If $g(m) = \frac{2}{m+1}$, find the range of the function $g(f(a))$.

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