
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
a) A piece of stovepipe made from a thin iron sheet is cut by two planes; one of them is perpendicular to the pipe's axis, and the other cuts it obliquely. The longest segment between the intersections is $c$, the shortest is $d$, and the diameter of the pipe is $2r$. Unfold the pipe along the segment of length $d$ and flatten it out. Provide a function whose graph matches the curved boundary line of the unfolded piece. What is the area of the metal piece?

b) Two vertical walls form a right angle. Each wall has a straight cylindrical cut-through with a diameter of $2r$, and their axes are at the same height and at a distance $e$ from the line where the walls intersect. To connect the two cut-throughs, two elbow pipes are made from cylindrical sections, one consisting of 4 segments and the other of 5 segments. The equally marked angles in the diagram are equal. Calculate the surface area of the two elbow pipes.

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