
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
The line $l_1$ passes through point M(-1, 0) and intersects the parabola $y^2=4x$ at points $P_1$ and $P_2$. Point P is the midpoint of the line segment $P_1P_2$, and the line $l_2$ passes through P and the focus F of the parabola. Let the slope of line $l_1$ be $k$.  
(1) Express the ratio of the slope of line $l_2$ to the slope of line $l_1$ as a function $f(k)$ of $k$.  
(2) Determine the domain of $f(k)$ and the interval where it is increasing.

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