
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 $\triangle ABC$, the equations of the lines containing sides $AB$ and $AC$ are $2x - y + 7 = 0$ and $x - y + 6 = 0$, respectively. Point $M(1, 6)$ lies on side $BC$.
1. If $AM$ is the altitude on side $BC$, find the equation of line $BC$.
2. If $AM$ is the median on side $BC$, find the area of $\triangle ABC$.

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