
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
$(1)$ Find the equation of the line passing through point $A(2,1)$ with intercepts on the $x$-axis and $y$-axis being equal.<br/>$(2)$ Given that the vertices of triangle $\triangle ABC$ are $A(2,1)$, the line containing the median $CM$ on side $AB$ is $2x+y-1=0$, and the line containing the altitude $BH$ on side $AC$ is $x-y=0$, find the equation of line $BC$.

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