
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 $\left\{ a_n \right\}$ and a geometric sequence with positive terms $\left\{ b_n \right\}$, where $a_1=b_1=1$ and $a_3+a_7=10$, $b_3=a_4$

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

$(2)$ If $c_n=a_n \cdot b_n$, find the sum of the first $n$ terms of the sequence $\left\{ c_n \right\}$, denoted as $T_n$

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