
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
Find all quadruples of positive integers  $(p, q, a, b)$ , where  $p$  and  $q$  are prime numbers and  $a > 1$ , such that   $$ p^a = 1 + 5q^b. $$  

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