
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
Point A (5, 1) is symmetric to point B $(x_1, y_1)$ about the x-axis, and symmetric to point C $(x_2, y_2)$ about the origin.  
(1) Find the equation of the line passing through the midpoints of sides BA and BC in $\triangle ABC$;  
(2) Calculate the area of $\triangle ABC$.

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