
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
Given a positive integer  $c$ , we construct a sequence of fractions  $a_1, a_2, a_3,...$  as follows: $\bullet$   $a_1 =\frac{c}{c+1} $  $\bullet$  to get  $a_n$ , we take  $a_{n-1}$  (in its most simplified form, with both the numerator and denominator chosen to be positive) and we add  $2$  to the numerator and  $3$  to the denominator. Then we simplify the result again as much as possible, with positive numerator and denominator.
For example, if we take  $c = 20$ , then  $a_1 =\frac{20}{21}$  and  $a_2 =\frac{22}{24} = \frac{11}{12}$  . Then we find that  $a_3 =\frac{13}{15}$  (which is already simplified) and  $a_4 =\frac{15}{18} =\frac{5}{6}$ .
(a) Let  $c = 10$ , hence  $a_1 =\frac{10}{11}$  . Determine the largest  $n$  for which a simplification is needed in the construction of  $a_n$ .
(b) Let  $c = 99$ , hence  $a_1 =\frac{99}{100}$  . Determine whether a simplification is needed somewhere in the sequence.
(c) Find two values of  $c$  for which in the first step of the construction of  $a_5$  (before simplification) the numerator and denominator are divisible by  $5$ .

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