
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
The problems are really difficult to find online, so here are the problems.

P1. It is known that two positive integers  $m$  and  $n$  satisfy  $10n - 9m = 7$  dan  $m \leq 2018$ . The number  $k = 20 - \frac{18m}{n}$  is a fraction in its simplest form.
a) Determine the smallest possible value of  $k$ .
b) If the denominator of the smallest value of  $k$  is (equal to some number)  $N$ , determine all positive factors of  $N$ .
c) On taking one factor out of all the mentioned positive factors of  $N$  above (specifically in problem b), determine the probability of taking a factor who is a multiple of 4.

I added this because my translation is a bit weird.
<details><summary>Indonesian Version</summary>Diketahui dua bilangan bulat positif  $m$  dan  $n$  dengan  $10n - 9m = 7$  dan  $m \leq 2018$ . Bilangan  $k = 20 - \frac{18m}{n}$  merupakan suatu pecahan sederhana.
a) Tentukan bilangan  $k$  terkecil yang mungkin.
b) Jika penyebut bilangan  $k$  terkecil tersebut adalah  $N$ , tentukan semua faktor positif dari  $N$ .
c) Pada pengambilan satu faktor dari faktor-faktor positif  $N$  di atas, tentukan peluang terambilnya satu faktor kelipatan 4.</details>

P2. Let the functions  $f, g : \mathbb{R} \to \mathbb{R}$  be given in the following graphs.
<details><summary>Graph Construction Notes</summary>I do not know asymptote, can you please help me draw the graphs? Here are its complete description:
For both graphs, draw only the X and Y-axes, do not draw grids. Denote each axis with  $X$  or  $Y$  depending on which line you are referring to, and on their intercepts, draw a small node (a circle) then mark their  $X$  or  $Y$  coordinates only (since their other coordinates are definitely 0).
Graph (1) is the function  $f$ , who is a quadratic function with -2 and 4 as its  $X$ -intercepts and 4 as its  $Y$ -intercept. You also put  $f$  right besides the curve you have, preferably just on the right-up direction of said curve.
Graph (2) is the function  $g$ , which is piecewise. For  $x \geq 0$ ,  $g(x) = \frac{1}{2}x - 2$ , whereas for  $x < 0$ ,  $g(x) = - x - 2$ . You also put  $g$  right besides the curve you have, on the lower right of the line, on approximately  $x = 2$ .</details>
Define the function  $g \circ f$  with  $(g \circ f)(x) = g(f(x))$  for all  $x \in D_f$  where  $D_f$  is the domain of  $f$ .
a) Draw the graph of the function  $g \circ f$ .
b) Determine all values of  $x$  so that  $-\frac{1}{2} \leq (g \circ f)(x) \leq 6$ .

P3. The quadrilateral  $ABCD$  has side lengths  $AB = BC = 4\sqrt{3}$  cm and  $CD = DA = 4$  cm. All four of its vertices lie on a circle. Calculate the area of quadrilateral  $ABCD$ .

P4. There exists positive integers  $x$  and  $y$ , with  $x < 100$  and  $y > 9$ . It is known that  $y = \frac{p}{777} x$ , where  $p$  is a 3-digit number whose number in its tens place is 5. Determine the number/quantity of all possible values of  $y$ .

P5. The 8-digit number  $\overline{abcdefgh}$  (the original problem does not have an overline, which I fixed) is arranged from the set  $\{1, 2, 3, 4, 5, 6, 7, 8\}$ . Such number satisfies  $a + c + e + g \geq b + d + f + h$ . Determine the quantity of different possible (such) numbers.



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