
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 prime  $p$ , let  $d(a,b)$  be the number of integers  $c$  such that  $1 \leq c < p$ , and the remainders when  $ac$  and  $bc$  are divided by  $p$  are both at most  $\frac{p}{3}$ . Determine the maximum value of \[\sqrt{\sum_{a=1}^{p-1}\sum_{b=1}^{p-1}d(a,b)(x_a + 1)(x_b + 1)} - \sqrt{\sum_{a=1}^{p-1}\sum_{b=1}^{p-1}d(a,b)x_ax_b}\] over all  $(p-1)$ -tuples  $(x_1,x_2,\ldots,x_{p-1})$  of real numbers.

*Brian Hamrick.*

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