
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 that the sum of the first $n$ terms of the sequence ${a_n}$ is $S_n$, and $S_n=n^2$ ($n\in\mathbb{N}^*$).
1. Find $a_n$;
2. The function $f(n)$ is defined as $$f(n)=\begin{cases} a_{n} & \text{, $n$ is odd} \\ f(\frac{n}{2}) & \text{, $n$ is even}\end{cases}$$, and $c_n=f(2^n+4)$ ($n\in\mathbb{N}^*$), find the sum of the first $n$ terms of the sequence ${c_n}$, denoted as $T_n$.
3. Let $\lambda$ be a real number, for any positive integers $m$, $n$, $k$ that satisfy $m+n=3k$ and $m\neq n$, the inequality $S_m+S_n>\lambda\cdot S_k$ always holds, find the maximum value of the real number $\lambda$.

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