
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) Let  $x>0,\ y$  be real numbers. For variable  $t$ , find the difference of Maximum and minimum value of the quadratic function  $f(t)=xt^2+yt$  in  $0\leq t\leq 1$ .

(2) Let  $S$  be the domain of the points  $(x,\ y)$  in the coordinate plane forming the following condition:

For  $x>0$  and all real numbers  $t$  with  $0\leq t\leq 1$  , there exists real number  $z$  for which  $0\leq xt^2+yt+z\leq 1$  .

Sketch the outline of  $S$ .

(3) Let  $V$  be the domain of the points  $(x,\ y,\ z) $  in the coordinate space forming the following condition:

For  $0\leq x\leq 1$  and for all real  numbers  $t$  with  $0\leq t\leq 1$ ,  $0\leq xt^2+yt+z\leq 1$  holds.

Find the volume of  $V$ .

*2011 Tokyo University entrance exam/Science, Problem 6*

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