
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
Let  $l,\ m$  be the tangent lines passing through the point  $A(a,\ a-1)$  on the line  $y=x-1$  and touch the parabola  $y=x^2$ .
Note that the slope of  $l$  is greater than that of  $m$ .

(1) Exress the slope of  $l$  in terms of  $a$ .

(2) Denote  $P,\ Q$  be the points of tangency of the lines  $l,\ m$  and the parabola  $y=x^2$ .
Find the minimum area of the part bounded by the line segment  $PQ$  and the parabola  $y=x^2$ .

(3) Find the minimum distance between the parabola  $y=x^2$  and the line  $y=x-1$ .

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