
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 increasing sequence $\{a_n\}$ that satisfies $a_{n+1}a_{n-1} = a_n^2$ (for $n \geq 2$, $n \in \mathbb{N}$), the sum of the first 10 terms equals 50, and the sum of the first 15 terms is 210, calculate the sum of the first 5 terms ($S_5$).

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