
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. Find all real numbers that satisfy the equation   $$ (1 + x^2 + x^4 + .... + x^{2014})(x^{2016} + 1) = 2016x^{2015} $$  p2. Let  $A$  be an integer and  $A = 2 + 20 + 201 + 2016 + 20162 + ... + \underbrace{20162016...2016}_{40\,\, digits}$ Find the last seven digits of  $A$ , in order from millions to units.


p3. In triangle  $ABC$ , points  $P$  and  $Q$  are on sides of  $BC$  so that the length of  $BP$  is equal to  $CQ$ ,  $\angle BAP = \angle CAQ$  and  $\angle APB$  is acute. Is triangle  $ABC$  isosceles? Write down your reasons.


p4. Ayu is about to open the suitcase but she forgets the key. The suitcase code consists of nine digits, namely four  $0$ s (zero) and five  $1$ s. Ayu remembers that no four consecutive numbers are the same. How many codes might have to try to make sure the suitcase is open?


p5. Fulan keeps  $100$  turkeys with the weight of the  $i$ -th turkey, being  $x_i$  for  $i\in\{1, 2, 3, ... , 100\}$ . The weight of the  $i$ -th turkey in grams is assumed to follow the function  $x_i(t) = S_it + 200 - i$  where  $t$  represents the time in days and  $S_i$  is the  $i$ -th term of an arithmetic sequence where the first term is a positive number  $a$  with a difference of  $b =\frac15$ . It is known that the average data on the weight of the hundred turkeys at  $t = a$  is  $150.5$  grams. Calculate the median weight of the turkey at time  $t = 20$  days.

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