
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 $\{a_n\}$ with the sum of its first $n$ terms as $s_n$, and $s_n = 2a_n - 2$ for $n = 1, 2, 3, \ldots$. If for the sequence $\{b_n\}$, $b_1 = 1$, and point $P(b_n, b_{n+1})$ lies on the line $y = x + 2$:

$(1)$ Find the general term formula for sequence $\{a_n\}$ and sequence $\{b_n\}$.

$(2)$ Let $c_n = a_n b_n$. Find the sum of the first $n$ terms of the sequence $\{c_n\}$, denoted $T_n$, and determine the largest integer $n$ such that $T_n < 167$.

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