
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
**Problem 1.** The longest side of a cyclic quadrilateral  $ABCD$  has length  $a$ , whereas the circumradius of  $\triangle{ACD}$  is of length 1. Determine the smallest of such  $a$ . For what quadrilateral  $ABCD$  results in  $a$  attaining its minimum?**Problem 2.** In a box, there are 4 balls, each numbered 1, 2, 3 and 4. Anggi takes an arbitrary ball, takes note of the number, and puts it back into the box. She does the procedure 4 times. Suppose the sum of the four numbers she took note of, is 12. What's the probability that, while doing the mentioned procedure, that she always takes the ball numbered 3?**Problem 3.** If  $\alpha, \beta$  and  $\gamma$  are the roots of  $x^3 - x - 1 = 0$ , determine the value of \[ \frac{1 + \alpha}{1 - \alpha} + \frac{1 + \beta}{1 - \beta} + \frac{1 + \gamma}{1 - \gamma}. \]**Problem 4.** The lengths of three sides,  $a, b, c$  with  $a \leq b \leq c$ , of a right triangle, are integers. Determine all triples  $(a, b, c)$  so that the value of the perimeter and the area such triangle(s) are equal to each other.**Problem 5.** Let  $A$  and  $B$  be two sets, each having consecutive natural numbers as their elements. The sum of the arithmetic mean of the elements of  $A$  and the arithmetic mean of the elements of  $B$  is 5002. If  $A \cap B = \{2005\}$ , then find the largest possible element of  $A \cup B$ .

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