
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
In a Cartesian coordinate system, vectors can be uniquely represented by ordered pairs of real numbers, allowing us to use vectors as a tool for studying analytic geometry. For instance, let's consider a line $l$ with inclination angle $\alpha$ ($\alpha \neq 90°$). Choose any two distinct points $P_1(x_1, y_1)$ and $P_2(x_2, y_2)$ on $l$. Assume that the direction of vector $\overrightarrow {P_{1}P_{2}}$ is upwards, then the coordinates of vector $\overrightarrow {P_{1}P_{2}}$ are $(x_2 - x_1, y_2 - y_1)$. Draw vector $\overrightarrow {OP}$ from the origin $O$ parallel to $\overrightarrow {P_{1}P_{2}}$, then the coordinates of point $P$ are $(x_2 - x_1, y_2 - y_1)$. The inclination angle of line $OP$ is also $\alpha$ ($\alpha \neq 90°$), and by definition of the tangent function, we have $k = \tan(\alpha) = \frac {y_{2} - y_{1}} {x_{2} - x_{1}}$. Study the following problems related to the line $Ax + By + C = 0$, where $(A, B, C \neq 0)$:
(1) Determine the relationship between vector $\overrightarrow{m} = (A, B)$ and the line $Ax + By + C = 0$, and explain your reasoning.
(2) If the lines $A_1x + B_1y + C_1 = 0$ and $A_2x + B_2y + C_2 = 0$ intersect, find the cosine of the angle between the two lines.
(3) Using vector knowledge, derive the distance formular from a point $P_0 (x_0, y_0)$ to the line $Ax + By + C = 0$, with $(A, B, C \neq 0)$.

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