
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 $\{a_{n}\}$ that satisfies ${a_{n+1}}=\left\{{\begin{array}{l}{{a_n}+2,n=2k-1,k∈{{N}^*}}\\{3{a_n}-2,n=2k,k∈{{N}^*}}\end{array}}\right.$, $a_{1}=2$, let $b_{n}=a_{2n}$.  
$(1)$ Write down $b_{1}$ and $b_{2}$, and find the general formula for the sequence $\{b_{n}\}$;  
$(2)$ Let $c_{n}=\log _{3}b_{n}$, find the sum of the first $10$ terms of $\{c_{n}\}$.

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