
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 \( a_{1}, a_{2}, a_{3}, a_{4}, a_{5}, a_{6}, a_{7}, a_{8} \) be positive integers such that
\[ a_{1}^{2}+\left(2 a_{2}\right)^{2}+\left(3 a_{3}\right)^{2}+\left(4 a_{4}\right)^{2}+\left(5 a_{5}\right)^{2}+\left(6 a_{6}\right)^{2}+\left(7 a_{7}\right)^{2}+\left(8 a_{8}\right)^{2}=204. \]

Find the value of \( a_{1}+a_{2}+a_{3}+a_{4}+a_{5}+a_{6}+a_{7}+a_{8} \).

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