
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
**p4** Define the sequence  ${a_n}$  as follows:
1)  $a_1 = -1$ , and
2) for all  $n \ge 2$ ,  $a_n = 1 + 2 + . . . + n - (n + 1)$ .
For example,  $a_3 = 1+2+3-4 = 2$ . Find the largest possible value of  $k$  such that  $a_k+a_{k+1} = a_{k+2}$ .**p5** The taxicab distance between two points  $(a, b)$  and  $(c, d)$  on the coordinate plane is  $|c-a|+|d-b|$ . Given that the taxicab distance between points  $A$  and  $B$  is  $8$  and that the length of  $AB$  is  $k$ , find the minimum possible value of  $k^2$ .**p6** For any two-digit positive integer  $\overline{AB}$ , let  $f(\overline{AB}) = \overline{AB}-A\cdot B$ , or in other words, the result of subtracting the product of its digits from the integer itself. For example,  $f(\overline{72}) = 72-7\cdot 2 = 58$ . Find the maximum possible  $n$  such that there exist distinct two-digit integers $ \overline{XY}$  and  $\overline{WZ}$  such that  $f(\overline{XY} ) = f(\overline{WZ}) = n$ .

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