
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
A sequence of the form $\left\{t_{1}, t_{2}, \ldots, t_{n}\right\}$ is called geometric if $t_{1}=a$, $t_{2}=a r$, $t_{3}=a r^{2}$, $\ldots$, $t_{n}=a r^{n-1}$. For example, $\{1,2,4,8,16\}$ and $\{1,-3,9,-27\}$ are both geometric sequences. In all three questions below, suppose $\left\{t_{1}, t_{2}, t_{3}, t_{4}, t_{5}\right\}$ is a geometric sequence of real numbers.

(a) If $t_{1}=3$ and $t_{2}=6$, determine the value of $t_{5}$.

(b) If $t_{2}=2$ and $t_{4}=8$, determine all possible values of $t_{5}$.

(c) If $t_{1}=32$ and $t_{5}=2$, determine all possible values of $t_{4}$.

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