
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
Let  $d$  be a real number. For each integer  $m \geq 0,$  define a sequence  $\left\{a_{m}(j)\right\}, j=0,1,2, \ldots$  by the condition
\begin{align*}
a_{m}(0)&=d / 2^{m},
a_{m}(j+1)&=\left(a_{m}(j)\right)^{2}+2 a_{m}(j), \quad j \geq 0.
\end{align*}
Evaluate  $\lim _{n \rightarrow \infty} a_{n}(n).$ 

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