
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]


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Given a sequence $\{a_n\}$ with the sum of its first $n$ terms denoted as $S_n$, $a_1=1$, and the point $P(a_n,S_n)$ (where $n \geqslant 1$ and $n \in \mathbb{N}^*$) lies on the line $4x-3y-1=0$. The sequence $\left\{\frac{1}{{b_n}}\right\}$ is an arithmetic sequence with the first term $-1$ and common difference $-2$.

$(1)$ Find the general formula for the sequences $\{a_n\}$ and $\{b_n\}$;

$(2)$ Let $c_n=\frac{1}{{a_nb_n}}$, find the sum of the first $n$ terms of the sequence $\{c_n\}$, denoted as $T_n$.

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