
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 $\{a_{n}\}$ where $a_{1}=a$, and $a_{n+1}=\left\{\begin{array}{l}{{a}_{n}+2,n=2k-1}\\{\frac{1}{2}a_n,n=2k}\end{array}\right.(k\in N^{*})$. When $a=1$, $a_{10}=$______; if all terms of the sequence $\{a_{n}\}$ take only a finite number of different values, then the values of all real numbers $a$ that satisfy the condition are ______.

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