
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
p1. It is known that  $m$  and  $n$  are two positive integer numbers consisting of four digits and three digits respectively. Both numbers contain the number  $4$  and the number  $5$ . The number  $59$  is a prime factor of  $m$ . The remainder of the division of  $n$  by  $38$  is  $ 1$ . If the difference between  $m$  and  $n$  is not more than  $2015$ . determine all possible pairs of numbers  $(m,n)$ . 


p2. It is known that the equation  $ax^2 + bx + c = $ 0 with  $a> 0$  has two different real roots and the equation  $ac^2x^4 + 2acdx^3 + (bc + ad^2) x^2 + bdx + c = 0$  has no real roots. Is it true that  $ad^2 + 2ad^2 <4bc + 16c^3$  ?


p3. A basketball competition consists of  $6$  teams. Each team carries a team flag that is mounted on a pole located on the edge of the match field. There are four locations and each location has five poles in a row. Pairs of flags at each location starting from the far right pole in sequence. If not all poles in each location must be flagged, determine as many possible flag arrangements.


p4. It is known that two intersecting circles  $L_1$  and  $L_2$  have centers at  $M$  and  $N$  respectively. The radii of the circles  $L_1$  and  $L_2$  are  $5$  units and  $6$  units respectively. The circle  $L_1$  passes through the point  $N$  and intersects the circle  $L_2$  at point  $P$  and at point  $Q$ . The point  $U$  lies on the circle  $L_2$  so that the line segment  $PU$  is a diameter of the circle  $L_2$ . The point  $T$  lies at the extension of the line segment  $PQ$  such that the area of ​​the quadrilateral  $QTUN$  is  $792/25$  units of area. Determine the length of the  $QT$ .


p5. An ice ball has an initial volume  $V_0$ . After  $n$  seconds ( $n$  is natural number), the volume of the ice ball becomes  $V_n$  and its surface area is  $L_n$ . The ice ball melts with a change in volume per second proportional to its surface area, i.e.  $V_n - V_{n+1} = a L_n$ , for every n, where a is a positive constant. It is also known that the ratio between the volume changes and the change of the radius per second is proportional to the area of ​​the property, that is  $\frac{V_n - V_{n+1}}{R_n - R_{n+1}}= k L_n$  , where  $k$  is a positive constant. If  $V_1=\frac{27}{64} V_0$  and the ice ball melts totally at exactly  $h$  seconds, determine the value of  $h$ .

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