
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
In the arithmetic sequence $\left\{ a_n \right\}$, $a_3=-5$, $a_8=10$, and $S_n$ is the sum of the first $n$ terms of the geometric sequence $\left\{ b_n \right\}$, with $b_1=1$, $4S_1$, $3S_2$, $2S_3$ forming an arithmetic sequence.

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

$(2)$ Let $c_n=|a_n|+b_n$, find the sum of the first $10$ terms of the sequence $\left\{ c_n \right\}$, denoted as $T_{10}$.

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