
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  $\Gamma$  be the maximum possible value of  $a+3b+9c$  among all triples  $(a,b,c)$  of positive real numbers such that
\[ \log_{30}(a+b+c) = \log_{8}(3a) = \log_{27} (3b) = \log_{125} (3c) .\]
If  $\Gamma = \frac{p}{q}$  where  $p$  and  $q$  are relatively prime positive integers, then find  $p+q$ .

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