
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. The range of a list of numbers is the difference between the largest number and the smallest number in the list. For example, the range of the list $1,5,1,6,3$ is $6-1=5$.
(a) Determine the range of the list $7,13,4,9,6$.
(b) The list $11,5, a, 13,10$ has a range of 12. Determine the two possible values of $a$.
(c) The list $6+2 x^{2}, 6+4 x^{2}, 6, 6+5 x^{2}$ has a range of 80. Determine the two possible values of $x$.
(d) The list $5 x+3 y, 0, x+y, 3 x+y$ has a range of 19. If $x$ and $y$ are integers with $x>0$ and $y>0$, determine the values of $x$ and $y$.

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