
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
<u>Set 2</u>**p4.**  $\vartriangle ABC$  is an isosceles triangle with  $AB = BC$ . Additionally, there is  $D$  on  $BC$  with  $AC = DA = BD = 1$ . Find the perimeter of  $\vartriangle ABC$ .**p5**. Let  $r$  be the positive solution to the equation  $100r^2 + 2r - 1 = 0$ . For an appropriate  $A$ , the infinite series  $Ar + Ar^2 + Ar^3 + Ar^4 +...$  has sum  $1$ . Find  $A$ .**p6.** Let  $N(k)$  denote the number of real solutions to the equation  $x^4 -x^2 = k$ . As  $k$  ranges from  $-\infty$   to  $\infty$ , the value of  $N(k)$  changes only a finite number of times. Write the sequence of values of  $N(k)$  as an ordered tuple (i.e. if  $N(k)$  went from  $1$  to  $3$  to  $2$ , you would write  $(1, 3, 2)$ ).


PS. You should use hide for answers.

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