
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
Let  $p,q$  be positive integers. For any  $a,b\in\mathbb{R}$  define the sets   $$ P(a)=\bigg\{a_n=a \ + \ n \ \cdot \ \frac{1}{p} : n\in\mathbb{N}\bigg\}\text{ and }Q(b)=\bigg\{b_n=b \ + \ n \ \cdot \ \frac{1}{q} : n\in\mathbb{N}\bigg\}. $$  The *distance* between  $P(a)$  and  $Q(b)$  is the minimum value of  $|x-y|$  as  $x\in P(a), y\in Q(b)$ . Find the maximum value of the distance between  $P(a)$  and  $Q(b)$  as  $a,b\in\mathbb{R}$ .

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