
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 parabola $y = ax^2 + bx + c$ ($a \neq 0$) that passes through point A $(1, 0)$, with its axis of symmetry being the equation $x = 3$ and its vertex being point B. A line $y = kx + m$ passes through points A and B, and the area of the triangle formed by this line and the coordinate axes is 2. Find the expressions for the linear function $y = kx + m$ and the quadratic function $y = ax^2 + bx + c$.

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