
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 a sequence ${a_n}$ where all terms are positive integers, for $n=1,2,3,...$, $a_{n+1}$ is defined as follows: $$a_{n+1}=\begin{cases} 3a_{n}+5,\text{ if } a_{n} \text{ is odd} \\ \frac{a_{n}}{2^{k}},\text{ if } a_{n} \text{ is even}\end{cases}$$ where $k$ is the smallest positive integer such that $a_{n+1}$ is odd.
(1) If $a_1=11$, find the value of $a_{2012}$.
(2) If there exists an $m\in\mathbb{N}^+$, such that $a_n$ is a constant $p$ for all $n>m$ and $a_n$ is odd, find the value of $p$.

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