
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 the sum of its first $n$ terms as $S_n$, $a_1 = -9$, $a_2$ is an integer, and for all $n \in \mathbb{N}^*$, it satisfies $S_n \geq S_5$.

1. Find the general term formula for $\{a_n\}$.
2. Let $b_1 = \frac{4}{3}$, and $b_{n+1}$ be defined as: if $n$ is odd, then $b_{n+1} = a_n$, and if $n$ is even, then $b_{n+1} = -b_n + (-2)^n$. Calculate the sum of the first $n$ terms $T_n$ of the sequence $\{b_n\}$.
3. Under the conditions of (2), consider a sequence $\{c_n\}$ such that $c_n = b_{2n} + b_{2n+1} + \lambda(-1)^n \left( \frac{1}{2} \right)^{a_n + 5}$ for all $n \in \mathbb{N}^{*}$. Determine if there exists a real number $\lambda$ such that $\{c_n\}$ is a monotonically increasing sequence. If it exists, determine the range of values of $\lambda$; if not, explain why.

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