
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 $\{a_n\}$ with $a_1=4$.
$(1)$ If $a_n=a_{n+1}+3$, find $a_{10}$;
$(2)$ If the sequence $\left\{ \frac{1}{a_n} \right\}$ forms an arithmetic sequence, and $a_6= \frac{1}{4}$, find the general formula for the sequence $\{a_n\}$.

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