
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 an arithmetic sequence $\{a_{n}\}$ with a non-zero common difference $d$, and the terms $a_1$, $a_3$, and $a_9$ form a geometric sequence, find the value of $\frac{a_1 + a_3 + a_9}{a_2 + a_4 + a_{10}}$.

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